Yes!

For example, just look at Neptune, with it's multiple moons.

It captured a new moon, Triton, and the whole system continues to function, even though Triton literally orbits in the opposite direction of all the other moons!

On another note:

If Triton had come screaming into the inner solar system, it would have certainly blazed like a comet, as all those volatiles, snows, etc, on its surface heated up, and then created a cometary tail unlike anything we humans have ever witnessed!

But if it spent considerable time captured in the inner solar system, then of course at some point it ceases being a "comet" when all the volatiles boil off, or sublimate away.

So when you say "comet" keep in mind, that when such objects cross over the frost line of the solar system, and enter the inner solar system, including Earth's region, then they'll only keep their "comet-tail" for so long.