I agree with the other answerer in that it is PARTIALLY true.
Specific jobs in the military branches are opening and closing all the time at different rates. The factors that govern this are many.
How overmanned or undermanned is the job?
How desirable or undesirable is the job? this affect the retention of those in the job and those seeking the job
How many does the service actually need?
In the CIC of a Navy ship we EWs (Now CTTs) occupied two watch positions. 6 of us were all that was need for a continual three section duty. OS's on the other hand occupied dozens of watchstations in CIC and as a consequence there were 30-40 or so OS's aboard ship but only 7 of us EWs.
Thus the OS school would be open more often than the EW school.
So how often a particular job is open varies from one job to the next and may open and close to recruitment multiple times throughout the year.
However... the end of a Fiscal Year and the start of an entirely new budget cycle can have an effect and it is possible that a particular school may have an opening then as any other time.
The only way to know for sure is to talk with a RECRUITER. Not just about which schools are available at the moment, but talk to them IN DEPTH regarding a specific Rating/MOS.
Recruiters will have access to information beyond just what is and isn't available at the moment. Ask them about retention and reenlistment stats for that job, ask about how often and how many billets are available for that job.
Talk to them about the job's CREO rating (A measure of how the job is at different ranks regarding staffing levels, whether it is overmanned, undermanned, or staffed just right.
Don't "Job Lock" yourself to a single job and only that job or else... a Recruiter will just ignore you and go on to someone else. But to a degree, you can hold out for a desirable job if you play it right and the job is not completely filled up and almost never opens.
If the job opens up on a semi-regular basis, just not at the moment. AND you are a very good catch as a recruit in all other aspects... the recruiter will work with you. Just don't be an *** about holding out for one job only or else.