I'm not too well versed in carbon composite or composite structures in general regarding fatigue, so I'll refrain from comment on that, but I know they tend to be damage intolerant and fail catastrophically.
Aluminum and other metals I know a little more about. It is typical, in any field of design, that the more specialized and higher tech the application and resulting solution, the more precisely the design is engineered for the specified requirements, and hence the less margin there is for "off-design" conditions, whether they be loading beyond the maximum specified overload, or use beyond the intended design life.
This is very typical in aerospace, where critically stressed elements of an airframe or aero engine have not only fatigue inspection intervals, but also maximum lifetime either in operating hours or number of cycles, beyond which they are no longer technically nor legally airworthy. There is no reason this should be any different for truly high tech bicycle components. Putting it another way, if a structural component does not have a maximum design lifetime, then it is not really that high tech. Nothing wrong with that, and that's why we C&V people value the old steel frames so highly.

A little more on fatigue, which is one of the two focal points of this discussion. Mike Mills gave a very good introduction to the subject. The point I want to make about steel, is that unlike most other metals, it has a "fatigue limit" (a counter-intuitive name) which really means if it never gets stressed above ~20% of its yield strength, it will never suffer any "cumulative damage". Rephrasing this slightly, steel ONLY experiences cumulative damage when it is stressed above ~20% of it's yield strength. Other metals, notably aluminum, experience cumulative damage every time stress is applied and removed, regardless of how low the stress (of course the lower the stress, the lower the damage). That is, in a nutshell, why aluminum bike frames are known to crack in fatigue, and why aerospace structural components are condemned after reaching their designed number of operating hours or cycles.

The other focal point of this discussion is "toughness", which translates to how much damage (i.e. deformation) a structural element can sustain before failing catastrophically. Again, steel is far and away the winner over aluminum and composite structures (in general, there can be exceptions), in part because it is lower tech and has inherently more design margin.