Should a bra cup cover the whole breast?
A bra cup shouldn't necessarily cover the whole breast, unless you're specifically wearing a full-coverage bra style. Other bra styles, like demi and perfect coverage, expose some breast tissue while still offering ample support.
However, the cup of the bra should never cut into your breast. If you're having spillage, that may indicate your bra cup is too small and you need to size up.