Some crystals can be described by a FCC lattice or BCC lattice. For example diamond can be described as a FCC lattice with two basis vectors. Is it also possible to describe it using an ordinary cube lattice with a different basis than the FCC basis?
Yes, it is possible. However, let me add some clarification:
A crystal is described by the combination of a lattice together with a motif. The lattice is independent of the physical crystal, it is a mathematical construct. Therefore, a priori, you could describe your crystal with several lattices, although there are some which are more convenient than others.
The motif are the coordinates of the physical atoms of the crystal in a given vector basis, so for the same crystal, depending on the chosen lattice, the corresponding motif coordinates will be different.
Finally, it is important to remark that no matter the lattice $\bigoplus$ motif combination which is chosen, one retrieves the same physical quantities (e.g. diffraction pattern) since the crystal is the same.