I'm following notations from sec. 14 Bott and Tu here:
Suppose $\pi :E\rightarrow M$ be a fibre bundle on a manifold with fibre $F$ and $\mathcal{U}=\{U_\alpha\}_\alpha$ be a good cover of $M$. Then consider the spectral sequence for the deRham-Cech double complex for $E$ w.r.t to the open cover $\{\pi^{-1}U_\alpha\}$. It has $E_1$ terms :
$E_1^{p,q}=\mathcal{C}^p(\mathcal{U},\mathscr{H}^q)$
where $\mathscr{H}^q$ is the presheaf w.r.t. $\mathcal{U}$ that assigns $H^q(\pi^{-1}U)$ to $U$.
Now this is a locally constant sheaf as $H^q(\pi^{-1}U)$ is isomorphic to $H^q(F)$. At this stage Bott and Tu say that this becomes a constant sheaf if $M$ is simply connected and so we'd have $H^*(E)=H^*(M)\otimes H^*(F)$ as vector spaces.
Now my question is : Isn't it possible to take a good cover of $M$ that is also a trivializing cover for $E$? Because if that was possible, our presheaf $\mathscr{H}^q$ would would become a constant sheaf right away and we'd have $H^*(E)=H^*(M)\otimes H^*(F)$ without $M$ having to be simply connected.
Can someone please clarify this ?
Yes. Consider the Klein bottle $S$ as an $S^1$-bundle over $S^1$.
If the formula was correct then we would have $b_{2}(S) =1$, but it is non-orientable so $b_{2}(S) =0$.
There are orientable examples as well, also given by torus bundles over the torus. I think the strongest possible result due to Serre when this kind of multiplicity occurs was that the fundamental group acts in a certain way on the fibre, which is clearly true when the base is simply connected. In general the strongest possible technique to compute the homology is the Serre sprectral sequence.