Can something travel faster than light if it has always been travelling faster than light?

Yes, if a particle would be travelling faster than light, it would always travel faster than light. This is what's called a tachyon, and they have in some sense imaginary mass.

The three regimes, time-like, light-like and space-like (i.e. subluminal, luminal and superluminal space-time distances) are invariant under Lorentz transformation. Therefore anything on a super-luminal 'mass-shell' would always stay there and could not be decelerated to light/ or sub-light speed.

The problem is not that it would violate relativity, but rather causality, since with faster than light information propagation one could 'travel back in time', therefore leading to paradoxes.

For an introduction check out Wikipedia