Can my bonus room support a 2000lb. antique pool table?

Not enough information to tell. We would need to know:
1. The size of the pool table, and how many legs it has.
2. The structure of the bonus room floor, meaning:
a. floor joist size
b. floor joist spacing
c. floor joist wood species and grade
d. floor joist span

However, assuming it's a full-size 5 foot by 10 foot pool table, that's 50 square feet. Your floor is probably designed for a "live" load of 30 pounds per square foot, so that's about 1500 pounds which is the maximum the floor can carry.

Will it collapse at 2000 pounds? No. Might the floor begin to sag somewhat? Maybe. Will the sag be noticeable? Possibly.

Assuming the bonus room is above a 2-car garage, there is probably a beam down the middle of the garage which supports your bonus room floor. If you can locate the pool table directly over this beam, and even better, directly over one or two columns which support the beam, the effects of the pool table will be mitigated.

So, the answer is, without seeing the actual conditions, probably YES, especially if the pool table can be located directly over the beam or columns below. But don't quote me )

The pool table in question is approximately 4x8 and has 4 legs and weighs 2000 lbs. The bonus room is 23 x 16. The floor joists are 16" I-beams on 16" centers. The span of the floor joists is 20.5'. Can my room support the pool table?
I hope this is enough info. Please let me know if it is not. Karen

Well,, now it depends on exactly which of the many types of composite joists ("I-beams") were used. The bad news is there's no center girder below. My seat-of-the-pants guess is that the pool table is a bit too heavy for the supporting structure. Your best bet is to have a competent architect or structural engineer examine your specific structure and make recommendations.