I have a tile roof. The clay tiles appear to be in good shape – at least not broken, however, I have several leaks. What would cause this?

The felt underlayment dictates the water tightness of the tile roof system.  Normally 3 plies of ASTM D226 heavy weight #30 felt is installed, as a minimum, under the clay tile roof system.  If the old felt is deteriorated the tile must be removed and the felt underlayment replaced then the existing tile may be relayed.