I recently purchased this (ridiculously expensive) Bluetooth module for my Arduino from SparkFun. On the item's page it says that it has been tested at 100 m. I contacted SparkFun for info about their setup and they said to achieve the 100 m range they used this 2.2 dBi antenna.
I presume any (2.4GHz) 2.2 dBi antenna will yield similar, 100m, results: is that correct?
However, I have this 7 dBi antenna on it's way. If I use that will I be able to get further range than from the 2.2 dBi antenna?
Yes, you will get more signal strength from a 7 dBI antenna than a 2.2dBI (specifically 4.8 dB). It solves that by radiating energy more directionally than an idea antenna that radiates evenly in all directions (0 dBI).
This increased signal strength of 4.8 dB is 10^(4.8/10) = 3 times more power. That will increase your range by about 70% in ideal conditions.
Since it is directional, you will need to point it more carefully. Specifically the linked antenna is pretty much a vertical wire. It radiates in a circle around the antenna; your receiver shouldn't be much above or below this plane.
You can think of antennas similar to your vision. 0dB would be considered you just as you with nothing artificial.
Now you decide that you would like to use a pair of binoculars to see further. The problem with binoculars is that your viewing range is not as large as you have with out them. However, binoculars are helpful, they let you see things that you couldn't see before. This is similar to lets say a 2.2dB antenna.
Now you decide you want to see even further, so you pull out a telescope. Again you are limiting the viewing angle, but it can be worth it in order to see further. This would be like a 7dB antenna.
Antennas are a little bit more complex, their baseline would be the ability to equally see in all direction (up, down, forward, backward, you name it) at the same time. This situation is called the isotropic antenna. This is where the 'i' in the dB comes from, and it is our baseline.
Going back to the example of binoculars and telescopes, antennas add a level of complexity to this because of this full 360* view that you start out with. You could have one antenna that has a pattern that still lets you see in-front, behind, to the left, and to the right but doesn't let you see above or below you. This type of antenna can have a gain because you cut out the above and below. Largely this would still be consider an omni-directional antenna because it still has a 360* view, but it wont be able to receive from directly above or below the antenna very well.
The basic concept that I am trying to get through is that gain can't just come out of no where, you have to sacrifice some part of the antenna pattern in order to give gain to another part of the antenna pattern.
So to your question of:
I presume any (2.4GHz) 2.2dBI antenna will yield similar, 100m, results
Not necessarily. Basically you could have a 2.2dBI antenna that has a really odd antenna pattern that causes you to have a lot of nulls in which you would have little range, while other areas might have a 100m range. To really find out you need to dig into the datasheet of the antennas.
It is worth noting that antenna manufactures will always do their best to try to make their antenna sound better than the competitors. This means they might measure their antenna gains in slightly different ways in order to get the biggest number possible. With any good antennas, you will be able to obtain proper antenna patterns.
The existing answers have mostly addressed your question, but just for posterity, I want to clarify a couple things.
You have to be careful with dBi, as it is not equivalent to total radiated power. Different antennas can have drastically different efficiencies.
What dBi tells you is the peak gain out of all possible directions when compared to a perfect antenna that radiates uniformly and omnidirectionally (isotropic). You should also note that this is a ratio, and that it is on the logarithmic scale, so 3 dB is 2 times more, whereas 20 dB is 100 times more (and the i in dBi means isotropic).
Anyway, the important thing to realize is that a 2.2 dBi antenna could have terrible gain in every direction except for what it is directly pointed at (a narrow beamwidth) and actually be radiating less total power than an omnidirectional antenna.*
When you are in line-of-sight (LOS) environments this peak gain is probably all that matters, as long as the antenna is in fact correctly pointed at the other antenna.** However, in indoor and non-line-of-sight (NLOS) environments, you can get a huge amount of multipath which will create crazy interference patterns -- the signal will bounce off the floors, ceilings, your refrigerator, your phone, etc., and depending on where you are these different reflections can add constructively or destructively, giving you drastically different received power. In these NLOS environments the efficiency of the antenna (total radiated power) often matters a lot more than the directivity (dBi).
* For example, a perfect 3 dBi antenna (2x gain) would radiate all of its power in 180 degrees, both azimuth and elevation (think half of a sphere). This is never achievable in reality, as it is always a gradual change in gain (notably, when you look at beam patterns they typically draw the 3 dB line, a heatmap would show a gradual change). However an antenna that achieved a 3 dBi gain in just an 18 degree beamwidth would also be considered a 3 dBi antenna, even though it is radiating 1/100th the power (since it is 1/10th as wide in azimuth and 1/10th as wide in elevation).
** In the absence of any other objects/reflections, the other antenna would only receive the power that was directly radiate towards it, so it really doesn't matter what the gain in any other direction is. Though, in reality, even with ground bounces you can get some screwy interference patterns.
Final thought -- if you look at a free space path loss calculator, e.g. https://www.pasternack.com/t-calculator-fspl.aspx, that 2.2 dBi gain gives you about a 22 m additional range (same pathloss at 78 m for a 0 dBi antenna as 100 m for a 2.2 dBi antenna). Your 7 dBi antenna would give another 75 m, up to 175 m for the same pathloss. Again, this is only in an ideal freespace (no reflections/absorption) and a perfectly pointed antenna.
You should also note that you can be breaking the law with too high a gain of antenna -- the FCC limits unlicensed transmission in the 2.4 GHz band to 1 watt EIRP (equivalent isotropic radiated power). Also, at some distance the bluetooth protocol will probably actually start to fail, as the latency from speed of light (about 1 us roundtrip at 175 m) can break things (though I'm much more familiar with WiFi).
Yes Isotropic means true "omni-directional. Since a rubber ducky, or patch antenna has null zones it sends more in some directions broadside to the antenna. THis gain typically 2~3dBi... accounts for the loss in other directions.
Highly directional TV antenna and satellite dishes start range often from 16~24 dBi. Gain and beamwidth of peak direction are tradeoffs from isotropic.
What this means to you is that when you are on the fringe, they can now aim to get 5dB more which is huge and that gets you into error free mode. But like a narrow beam headlight, it also means if are far away and do not know the direction to the router or cell tower, you are more likely to get lost until you monitor your RSSI or received signal strength indicator on the cell phone. For Wifi however, it serves a dual purpose. Once a connection is made it reverts to baud rate and not signal strength in some cases such as Apple's OSX, and if you lose the signal , then you have to aim to maintain a good connection.
For direct ideal "interference-free" point to point in clear line of site, 5 dB improvement means you can nearly double your distance. This rarely happens in the city, so distance is not as significant as ability to aim towards signal and away from interference.
If one wanted to calculate path loss they might use the "Friis Transmission Equation" for path loss. 
This does not account for receiver noise floor, multi-path dead zones and path loss from buildings, trees, rain etc.
Range, R is in meters, as is Lambda the wavelength of transmitter Ft and gains for both antenna Gr, Gt.
I presume any (2.4GHz) 2.2dBI antenna will yield similar, 100m, results
No. I'm not an expert on antennas, but I've heard about directive antennas. The "i" in dBi stands for "isotropic", meaning uniformly radiating in all directions. Such an antenna doesn't really exist, but the theoretic model can be used as a reference. So a 2.2 dBi antenna does 2.2 dB better than the isotropic antenna.
Saying any 2.2 dBi antenna will yield the same distance ignores the antenna's directivity. An antenna with higher directivity will achieve the 100 m with less power than a less directive antenna.
