Definitely. There are lots of people who play all of the woodwinds (maybe not bassoon) very well; modern musicals almost exclusively call for this kind of doubling. In fact, it's kind of unusual to find professional saxophonists or clarinetists that aren't at least competent at the other, though admittedly those instruments are remarkably similar.
And it's not uncommon to find people who play multiple brass instruments, especially the more closely related they are. Trombone+euphonium is very common and can just about be expected of a pro, and trumpet+horn or trumpet+euphonium (or all 3) is pretty common too. These doublings are far less commonly asked for in any kind of published music, but the instruments are fundamentally the same so it's really no surprise.
Crossing over between the woodwind and brass families is rarer, but still not unheard of. The sound production mechanisms start to vary far more so there's less direct skill transfer. In particular, brass players often struggle with flute because the total lack of resistance causes them to far overblow. But it's nothing that good ol' practice can't overcome.
My advice for getting used to the switch is very simple: just keep practicing the new instrument, you have to give your body enough time to internalize the muscle memory and learn to make the change faster. I started on trumpet and when I first started learning horn, one would totally ruin the embouchure for the other, but now I can switch between them as quickly as I can get them to my face. I've never tried quick switching between brass and woodwinds but I don't imagine having too much of a problem.