Is it normal for your tampon to feel like it's starting to slip down a teeny bit?

Depends: It depends on the size of the tampon and the flow of the menses. Some women with a very heavy flow may have some slippage. If is not uncommon for those who have slippage to use the next larger size or to wear a pad in addition to the tampon. If it continues to be a problem consult your md.

Possibly: Make sure you are applying proper technique with a nurse or OB-Gyne to be sure