Westhawk: Those are valid questions about adequate versus legal crew rest. It 'allegedly' makes no difference at all to some of our 'friends' with the FAA (check the accident at LIT which 'motivated' the FAA to give standby crews a daily period of rest...), or to many of our management personnel. How little do many of the "suits" realize that an accident might seriously affect the stock price...and are lawsuits getting much cheaper for corporations?
Different Vector: That is a common scenario. And when we also are tired from a lack of sleep, it is much more difficult to be sure that we are run down by a common cold etc.
Also: you definitely don't want to report to a briefing room for a check ride if this could be the case.
It can really relax your decision making abilities, a bit like a little hypoxia. They can find a seat-warmer pilot somewhere in the building, such as the superb, good-looking lady from reindeer country.
If you are flying a real trip, call Crew Sched. before you takeoff and let them replace you at the hub after this leg, or if need be, at a spoke airport. Even if you are fortunate enough to fly in a 3-person c0ckp1t.