A famous result by Thom states that Oriented Bordism classes are determined by characteristic numbers; specifically, two closed manifolds are orientedly bordant if and only if they have the same Stiefel-Whitney and Pontryagin numbers (I'll just talk about Stiefel-Whitney for brevity). An immediate consequence is that if $M$ is a closed manifold which has a non-vanishing Stiefel-Whitney number involving $w_k$ for some $k$, then $w_k(N)\neq 0$ for any $N$ which is bordant to $M$; in other words this non-vanishing characteristic number provides a "bordism reason" for why a characteristic class should be non-zero.

My question concerns the converse. Given an $M$, suppose that for some $k$ every Steifel-Whitney number involving $w_k$ vanishes, so that there is "no bordism reason" for the class to be non-vanishing: is it then possible to find an $N$ which is bordant to $M$ and has $w_k(N)=0$? If so, is it possible to simultaneously eliminate *all* classes which have no bordism reason to be non-zero?

This seems like something which maybe shouldn't be expected since characteristic classes often provide obstructions to doing surgery, and two manifolds are orientedly bordant exactly when they differ by a finite sequence of surgeries. On the other hand the manifold $\mathbb{CP}^n\#\overline{\mathbb{CP}^n}$ is null-bordant even though it has many non-vanishing characteristic classes for $n>1$ (half the time it isn't even spin).