Orientable manifolds have even Euler characteristic unless the dimension is a multiple of 4. I give a generalisation of this theorem: $k$-orientable manifolds have even Euler characteristic (and in fact vanishing top Wu class), unless their dimension is $2^{k+1}m$ for some integer $m$. Here we call a manifold $k$-orientable if the $i^{\rm{th}}$ Stiefel-Whitney class vanishes for all 0 < $i$ < $2^k$. This theorem is strict for $k=0,1,2,3$, but whether there exist 4-orientable manifolds with an odd Euler characteristic is a new open question. Such manifolds would have dimensions that are a multiple of 32. I discuss manifolds of dimension high powers of 2 and present the results of calculations on the cohomology of the second Rosenfeld plane, a special 64-dimensional manifold with odd Euler characteristic.