If a lot of dust and other such filth gets into them, they can eventually overheat by way of insulation. But catch fire? Highly unlikely, especially with a transformer rated so low and with so few winds around the iron. Melt? Yeah, it can happen, but the chance of that happening is quite low. In order to catch fire, the transformer would have to have a very clean short circuit on the AC side and somehow not trip the line breaker or fuse, and then sit there shorted for a good 5 seconds or more (enough time to build up enough energy to cause actual damage)...way longer than any breaker or fuse can maintain connection.
I used to do my own testing like this when I was younger (much to my dad's dismay...he was an electrician so he had all the fun stuff). A normal 10A fuse will blow in less than 1/8 of a second when the circuit is shorted. Higher amperage fuses seem to take slightly longer; it took about 1/4 second to blow a 30A fuse. The typical breaker will take slightly longer to cut the circuit than a fuse, but even a 50A breaker is going to break in less than a half a second with a clean short. Needless to say, the circuit is going to be cut long before the transformer is able to catch fire by way of short circuit on the AC side...and even then, that's provided that the heat generated doesn't sever the connection on its own...as the wires heat up, the metal becomes fragile, and the wiring in a transformer is extremely thin...it would likely sever itself in a fraction of a second anyways. If that ever happens, the transformer simply stops working because there is no longer a working electric field.
Bottom line: whoever told you that was full of pockeysauce and doesn't know the first damn thing about electricity.