We all easily understand the problems of under tightening a screw or bolt - if is too loose, things move, it ends badly and we know why. Most of us also know that too much torque can damage the components, and in extreme cases strip the thread of the fastener. What many people do not understand is that bolts stretch when torqued. At the correct torque the bolt is elastic or springy, and is this elasticity that holds things together under vibration.
An over torqued fastener can be stretched too far and loses elasticity long before it breaks or threads strip - at this point, the bolt may not continue to hold the surfaces tight under vibration, and can work loose. As the bolt has been stretched beyond its elastic ability, it will not return to its original size and essentially stuffed - however you cannot see this visually. Here in lies a big problem - it came lose, things come lose because they were too lose, so people assume it needs to be put on, just tighter. Unfortunately, that over tightened bolt (which is now stuffed) will only hold if it is over tightened even more.
On cheap bikes with lots of steel (very forgiving material) and excess weight, its never a real issue - you can give a gorilla a power bar and not do much damage. The use of things like titanium and aluminium rather than steel for fasteners, along with the desire for weight savings means these components are designed with fine tolerances - hence its easy to over do it and the correct torque becomes critical.
Essentially if you bike is not a chain store special, over torquing will damage something - hopefully just the bolt that is easily and cheaply replaced. If you torque a bolt up to specification and it does not hold reliably, replace the bolt - no matter how tempting, do not over tighten it.