How did the socket fail? RCD only looks for (trying to explain in simple terms - it's not quite like this!) a difference in the electricity going into the appliance, or the circuit in the house, and the amount of electricity coming back to the fusebox.
If more is going out from the fusebox (or socket if that's where the RCD is) than is coming back then it trips as it assumes some of the electricity is going astray. Through a fault somewhere (or in worst case, through a person!).
So only certain types of failures will cause an RCD to trip. An appliance can sit burning through 13A of current and catch fire easily (think how hot an electric fire can get) without tripping anything, as that's seen as normal to the protective devices (the reason just sticking a 13A fuse in everything is a really bad idea - most things don't need that).
If you are worried the RCD isn't working, do you regulalry test it? It'll have a test button on it.
If you don't have a test button then it's not actually an RCD - maybe it's a MCB - miniture circuit breaker. This is basically a clever fuse, and trips when too much electricity is flowing. In a fuse box, on sockets it'll often be 32A. That's more than enough for serious issues, so they don't always trip!
Test the RCD, and it it trips it's probably fine. If you are really worried, get an electrician to test it properly. Adding an RCD socket isn't always wise if you have one at the fusebox as well, as having two on the same circuit can cause issues (confusion as to which will trip first etc). It also won't help unless rated as more sensitive than the main one - talk to an electrician.
Hope that helps a bit - I've probably not explained it very clearly!