I can get a tax break if I was a student for 5 months. I graduated in the middle of May, so I wasn't a full-time student for all of the month. Can I still get tax break?

You would still be considered a student for the education credits.

That requirement is at least half time at least one academic period.

Additionally it would fulfill the requirement of being a student for someone else to claim you as a dependent (at least part of at least 5 months) but for someone to claim you, there are other requirements as well. 

You do meet that rule. But, you do not need to have met the 5 month, full time student rule to claim a tuition credit.  That rule is primarily used to determine if you can still be claimed as a "Qualifying child" dependent  by others.