The answer is yes.
There can be $f\in W^{1,p}(\Omega)\cap C(\Omega) \cap L^{\infty}(\Omega)$ such that $f=0$ on $\partial \Omega$ but $f\notin W^{1,p}_{0}(\Omega)$.
If $f\notin C({\bar{\Omega}})$, then its trace on $\partial \Omega$ can be different from zero, and in that case $f\notin W^{1,p}_{0}(\Omega)$.
Example: simply the characteristic function $\chi_{\Omega}$