Carbon bars, cranks, stems etc? Absolutely not.
I don't have a problem with these parts being carbon as long as they are:
a. of top-notch build quality; and
b. haven't been crashed.
The former is fairly easy to tell if it's legitimately from a major carbon manufacturer. And even some of the no-name stuff is made in factories that make a lot of carbon parts. I have a no-name carbon post on my Moots and it's really well made and very strong (had a framebuilder look at it and he was duly impressed with the quality, but it can be hit-or-miss).
The latter is the bigger worry. If you use carbon handlebars, damage can be hidden by grips, bar tape, or control mounts. And when carbon bars fail, they tend to fail in a very binary way. Then again, so do aluminum bars, though they'll typically bend before they snap.
Carbon stems are typically out of harm's way in the grand scheme of things, and all but the most expensive ones use alloy for the contact points with handlebar and steering tube.
On cranks the liability is any inlaid metal part. I've known the aluminum or steel pedal thread inserts in carbon cranks (e.g. SRAM Red22) to come loose over time.
And that leads to cases where carbon is mated to metal. In the case of mating to steel or titanium, it's no big deal and tends to be a strong connection. With aluminum it's different, and said join points need to be checked regularly for delamination and oxidization, both of which can lead to joint failure.
So I'd steer clear, if possible, from things like carbon rear triangles that are epoxied to aluminum front triangles, or carbon bladed forks with aluminum steering tubes. I've seen failure (or pre-failure) on these a few too many times to be convinced it's a great solution.
And with any carbon part that gets clamped, it's
absolutely necessary to use anti-slip paste and a torque wrench. The former will keep things from, well, slipping, as well as allow you to tighten bolts on the lower end of the recommended torque spectrum (read: less likely that you'll compromise carbon layering). And the torque wrench - well, that just makes sense, right? If you own a carbon frame or any carbon components that require bolts to be tightened, get one and treat it properly (e.g. wind it back to zero after every use).