Yes, there is a failure-to-launch rate associated with any missile. This is one of the reasons for having more missiles than most people think are "necessary" — a redundancy effect.

It is worth noting that the missiles that are most at risk for this kind of thing are liquid-fueled. The missiles used on submarines and even many land-based missiles are solid-fueled. While it is possible to have accidents with solid-fueled rockets, they are on the whole more reliable and easier to handle, because their fuel simply cannot all ignite at once (whereas if the oxidizer tank collapses into the fuel tank of a large liquid-fueled missile, the entire thing will go up — this happened at the Damascus accident in 1983). There are tradeoffs to using solid fuel so they are not universally adopted.

Having the missile body explode would not necessarily, and not likely, result in the warhead detonating. During the aforementioned Damascus accident, the warhead was launched into the air and landed nearby without significant damage. Other accidents in the earlier Cold War did result in the spreading of plutonium in the local area. This would not be ideal under wartime conditions, obviously, but it is not the same thing as the entire place getting nuked. The heavy silos used for some of the missile bases (e.g. the Titan silos) were built to be able to sustain the accidental ignition of the missile fuel.

Eric Schlosser's Command and Control is a great read about nuclear weapons safety, and spend about half the book dissecting the Damascus accident.