What are the maximum levels in Fallout Shelter?

Your dwellers can level up until Level 50. Each level of your dweller will increase their health points.

The S.P.E.C.I.A.L. stats will be capped at 10 for training. If you reach a value of 10 at a S.P.E.C.I.A.L. attribute you won't be able to train the dweller further in this stat. Instead of further training, you are able to give them better cloths which can raise their S.P.E.C.I.A.L. attributes further. A good tactic is to train your dwellers in their main stat until 10 and probably their Luck too (as it will increase your rush-chance). After that you can provide a Luck-based or a working specific outfit to your dweller.