I'm certainly not an expert at this kind of problem, but, here is one example, which is found in K.Grove-W.Ziller, Lifting group actions and nonnegative curvature, Trans. Amer. Math. Soc. 363 (2011) 2865-2890. (And here is a link to a free version).
(Grove-Ziller) Suppose $E$ is a vector bundle over $\mathbb{C}P^2$ with non-vanishing second Steifel-Whitney class. Then $E$ admits a complete metric of non-negative curvature.
(They have similar results over all closed simply connected $4$-manifolds known to admit metrics of non-negative curvature: $S^4$, $\mathbb{C}P^2$, $S^2\times S^2$, and $\mathbb{C}P^2 \# \pm \mathbb{C}P^2$).
I know, you wanted an example using principal bundles, not vector bundles. This comes in the proof.
The idea is that every vector bundle can be written as an associated bundle $E = V\times_{O(n)} P$ for some principal $O(n)$ bundle $P$. (In fact, because each of the $M$s above is simply connected, one can use $SO(n)$ instead).
Since the usual metric on $V\cong \mathbb{R}^n$ is $O(n)$ invariant and non-negatively curved, the O'Neill formulas for a Riemannian submersion imply that if you can show $P$ has an $O(n)$ invariant metric of non-negative curvature, so does $E$.
So, how does one show that $P$ admits non-negative sectional curvature? Because $P$ is compact, it has a chance of being "built" out of compact Lie groups in nice ways - more specifically, $P$ has a chance of belonging to the class of compact homogeneous spaces, compact biquotients (quotients of homogeneous spaces by free isometric actions), or compact cohomogeneity one manifolds (spaces with a Lie group acting with codimension 1 orbits).
It turns out that many such $P$ are cohomogeneity one manifolds, which, by previous work of Grove and Ziller (K.Grove-W.Ziller, Curvature and Symmetry of Milnor Spheres, Annals of Mathematics 152 (2000),331-367.) admit metrics of non-negative curvature.
The characteristic classes enter in the following way. The Grove-Ziller construction allows them to construct many $SO(n)$ principal bundles $P\rightarrow M$, but a priori, it's not clear which bundles are actually constructable in this manner. However, their construction allows the to compute the characteristic classes of the tangent bundle to $P$, which in turn allows them to compute the characteristic classes of the principal bundle $P\rightarrow M$ itself. Finally, Dold and Whitney (A. Dold and H. Whitney, Classification of oriented sphere bundles over a 4-complex, Ann. of Math.
(2) 69 (1959), 667-677.) have already shown that the characteristic classes of an $SO(n)$ principal bundle over a $4$ manifold completely classify the bundle. This allows Grove and Ziller to recognize that, for example, they have all $P\rightarrow \mathbb{C}P^2$ with non-vanishing second Stiefel-Whitney class.
