The middle of January isn't a prime moving time for FSU students – it's true. That said, there are many reasons that you may be looking for an apartment near FSU at really any time of the year. And because of that, it is very unlikely that you will be in a situation where there are no FSU apartments available right now.
Some students move during the middle of the year due to roommate issues. Sometimes there are just people who you can't live with. (And there's nothing wrong with that.) Sometimes, you graduate at an ‘off' time. Or, perhaps you are not starting as a traditional student straight out of high school. Some students may want to take some time to get settled into Tallahassee before beginning classes.
While most student housing is rented out around August, most places end up with some vacancies throughout the year. Sometimes people leave, and sometimes a complex has had an empty unit for months that they would love to rent out. You may be able to find a room in a place that rents out by the room (individual leases) rather than by the unit. This way, you would already be set up with roommates.
As January is the start of the Spring semester, you may be able to find a vacancy left by someone who graduated or dropped out in December. This will not be as true next month, but if you need an FSU apartment immediately, you may be able to find a place under these circumstances. If you find a unit that has individual leases, you will probably end up with roommates that already have a set pattern and agreements for living together. This may or may not work out well for you – it may be advisable to ask to meet the roommates first.
Starting a lease at this time of year may give you a chance at a better deal on a FSU apartment. Any unit not rented at this point is much less likely to be rented now than in August or September. Thus you may get a better deal, but your landlord may only offer you a lease until about August, at which point you would have the option to renew your lease.
If you do end up with a lease that ends in August, figure out quickly if you will want to renew or move at that time. Many students begin looking for their next home during the Spring semester. While it may be hard to know so early what you will be doing months from now, it's best to secure your next year's rental as soon as possible.