One seeming problem with the third rail system would be the loss of power that a train would experience when it crosses switches or otherwise loses contact with the third rail. However, this problem is easily rectified. The CTA solves this problem by having a battery system in each car. Thus, as the train passes across the gap between two parts of the third rail, the battery takes over, keeping the lights on. The only noticeable change for passengers inside the train is the fact that the blowers for the heating or air conditioning switch off temporarily. It is only if a train is stopped over a gap in the third rail, or if, on a curve, the train loses contact with the rail for an extended period of time, that the lights will shut off.
Here in Chicago, there are tools called "stingers" to help trains stranded in just these types of situations. The stingers are basically live wires with insulated handles that can be touched to the contact shoe on the train to give it a momentary burst of power, enough to get the train back onto the third rail. Stingers are seen at common lost-power points, such as switches, sharp curves, and level crossings.
Another complaint about third rails is the danger of electrocution involved in using them. However, this is not as much of a problem as many may think. First off, transit systems put a lot of work into keeping people off of the tracks, regardless of whether or not a third rail is used. Anyone who does manage to make it on to the tracks also puts himself at risk of being hit by a fast-moving train. Secondly, on elevated train lines, it is very hard to electrocute yourself on the third rail. This is because the rail sits on glass insulators, and the tracks themselves sit on wooden ties. Thus, someone touching the third rail doesn't get electrocuted because she isn't grounded. However, if someone touches both the third rail and one of the remaining two rails, closing the circuit, or manages to ground himself, then he will be electrocuted.
Rain can also help to close a circuit in a third rail system. Normally, the ceramic insulators and wooden ties are enough so there is no danger of the system shorting out in any way. However, the constant rubbing of contact shoes and wheels on the rails causes small bits of rust to come off. Some of this dust settles on the glass insulators and wooden ties, and eventually these get coated with a light layer of grime and rust. This isn't a problem normally, as the rust-grime combination is a poor conductor, but when it rains, the water is enough to cause a circuit to form from the third rail, through the rusty grime on the insulators and cross ties, and down the metal superstructure to the ground, or in the case of grade level tracks, through the grime directly to the ground. The result is a very low-grade short across the entire line. Thus, it isn't uncommon to see CTA workers standing or sitting on the third rail when it's dry (on the elevated tracks only, of course), but as soon as the first drops of rain fall, the workers rush off of the tracks.