Yes, typically. For D1, any student has 5 calendar years from his first full-time collegiate enrollment in which to use his 4 seasons of eligibility for competition in any particular sport. So playing one sport for 4 years (using up all eligibility in that sport) doesn't stop a student from playing a different sport.
The above also applies to a player who is a graduate or post-bacculaureate student at the original college. However, if a player transfers schools, the transfer rules come into effect.
Happily, for most sports, there is a one-time exception to the year of residency requirement, and the graduate student at a new D1 institution would be eligible. Note that it doesn't matter if the original school was NAIA or D3 or D1, or didn't have any affiliation at all. It just matters whether the player has transfered.
So a baseball player can play 4 years at one school, graduate, transfer to a D1 school, and play a different sport for the remaining year in the 5 year period-- assuming he didn't have a previous 4-4 transfer.
It usually won't work the other way around: If a track athlete goes to graduate school at a new D1, he can't play baseball unless he never received any athletic money, and wasn't recruited for baseball (I think) at the original school
The main relevant D1 Bylaw is 14.1.9.
For D2 and D3, the rules are somewhat different, e.g. 10 full-time semesters rather than 5 years, and the transfer rules are less restrictive. Typically competing in a different sport in the 5th year (or later year if college attendence was interrupted) would be within the rules.
