Yesterday folks from my ISP came to install a 16 Mbps "Fiber to the Home" connection, but I noticed that the connection still comes in through the same two wires on our phone jack.
Could that same connection handle a 25 Mbps connection? What about 100 Mbps? Is there a no theoretical maximum, but it becomes more and more unreliable?
There's one beautiful, simple equation that has it all, called the Shannon–Hartley theorem:
$$C = B \cdot log_{2}\Big(1+ \frac{S}{N}\Big)$$
It says that over a channel with a given quality, the capacity (bit rate) C is proportional to the channel's bandwith B. The quality of the channel (signal vs. noise) hides within \$log_{2}(1+S/N)\$, and the bit rate includes redundancy (error checksums and the like).
The best data rate can be achieved with a low-noise installation of a channel that offers a high bandwith.
Of the wiring systems in question, a simple two-wire phone line will have the lowest bandwith and the worst noise properties (crosstalk and interference from neighboring lines, ...), twisted pair wires increase the bandwith and are more immune to external noise with an increasing "CAT number" (6 being better than 5e, being better than 5, ...) and systems with optical fibers are even better.
A telephone connection has an audio bandwith limited to a few kHz. Old systems had filters and the wires were often not capable of much more than the filter-defined bandwith. Digital subscriber lines (DSL) take advantage of the fact that many phone lines, when not being filtered, can take more than the mediocre audio bandwith of telephones. Beyond approx. 200 bps, it depends on the installation of the last mile and in your house (and your provider's willingness to use it in the best way). Typically, fiber can handle a greater bandwith than copper, but good quality can be achieved with copper, too.
Note: Someone selling you "Fiber to the Home" over copper wires is just doing (un-? clever?) marketing. Claude Shannon was way cooler, he didn't even care about the type of channel (copper, fiber, radio waves, whatever), he just looked at the bandwidth and the quality (signal-to-noise ratio). You can join Shannon and, like him, enjoy the theory and also don't care about the material of your wires. When I took my communication theory class at college, my professor was actually very right when he pointed out the beauty of shannon's work and said that the equation mentioned above was the E = mc2 of the information technology age.
The other answers hit on some good points, but they are not all 100% correct. If they advertise fiber to the home then they must actually be running fiber to your property. They can convert it to copper at that point, but if they don't get the fiber to your property then they can't be calling it "Fiber to the Home".
History
What originally caused phone lines to be slow for data communications was the low pass filter that was placed on the phone line. zebonaut is correct in his definition of channel capacity. It has been awhile since I have messed with that stuff, but I believe the low pass filter was set around 8 KHz.
DSL then came around and took advantage of the fact that the phone companies had physical access to the lines before the low pass filter. This meant that they could use frequencies that voice traffic didn't exist in but that the copper could transmit. There was still low pass filters put in place for voice traffic so that you wouldn't here high pitched noise on your phone call or have any aliasing problems.
DSL was limited in speeds based off of the S/N portion of zebonaut's channel capacity formula. The longer the wire was the worse SNR you would get. As time as gone on phone companies have continually shortened the copper loop and replaced old copper with fiber. As this has happened DSL speeds have been able to increase.
It has gotten to the extent in many areas that there is fiber almost all of the way to the house. This as made it so that the cost to get fiber all of the way the property is actually rather reasonable. This also provides the ability to advertise "Fiber to the Home" and like I said earlier they can't call it that unless it actually is that.
One example of a situation of not having "Fiber to the Home" is AT&T's U-verse. In many cases they actually do have fiber to the home, but because they don't have that everywhere they don't advertise "Fiber to the Home", but rather they advertise "fiber optic technology and computer networking".
Fiber versus Copper
One of the big reasons we use fiber instead of copper is how much noise copper can pick up where as fiber picks up almost no noise. This makes the SNR very good which allows for much higher data rates.
By getting fiber all of the way to the property, the length of copper needed is minimized as well as there are no neighboring copper lines that can cross talk into your line.
So basically, the less copper you have the higher the data rate you can get.
The issue here is not simply how fast copper wire can transmit data. The issue is how fast can it transmit over a specific distance. As mentioned in the comments, Ethernet can transmit at 1GBps or more, but only for around 100 meters (I don't recall the actual distance). CAT6 cable (used by Ethernet) has 8 wires, grouped into 4 pairs: 2 pairs are unused, one pair for transmit, and one pair for receive. Each of the pairs of wires are twisted at a different rate to reduce cross-talk between pairs. The wires are paired because over long distances (and/or high speeds), you can't just compare a single wire's voltage against a common ground wire because ground may be noisy, etc.). Instead, you compare the difference between the two wires.
So, back to your new internet connection: "Fiber to the Home" is a Marketing department's way to compete with other companies that actually do run fiber to the home in some areas. What they're really doing is running fiber to the local "central office", and then running a copper pair for the "last mile". They market this as new, but they've been running fiber between central offices for years (everyone does it). The change is that they eliminate the telephone equipment at both ends and use new protocols to handle higher speeds (that's why DSL has always been handycapped: it has to share the line with existing telephone equipment).
A bigger problem then what copper can theoretically handle, is the fact that most "last mile" copper was laid decades ago, so it's not twisted like CAT6 (or it's twisted to reduce noise on voice traffic, not high speed data traffic), it may be corroded, and the wiring at both ends may be bad (you should've seen the wiring in my parent's 1929 house!).
