My tub feels like its caving in the center of tube (bottom), Is it normal for tubes to do that? Do I have to get it checked?

If this is a steel or vinyl tub it should be bedded in mortar to keep it from deflecting. If not u can eventually develop a leak at drain from the movement.

If u can get under tub from back, it is not to late to do it now.

Chuck

Other than simply replacing the tub and making sure the new one is supported properly, it's not an easy repair. I did have that same experience once though and repaired it successfully. I cut a long, low access hole in the wall of the adjoining room so I could gain access to the underside of the tub. I then took fiberglass cloth, soaked it in resin, bunched it up and jammed it in under the tub to reinforce the support in the "spongy" areas. Once the resin cured the tub was solid as a rock! I then put the piece of drywall back in place, taped and Spackled the wall. That was at least 15 years ago and the tub is still fine. It's a bit messy so if you do it, make sure you put a tarp down in the work area. Good luck!