Atiyah and Hirzebruch proved that if a compact connected Lie group acts smoothly and nontrivially on a compact spin manifold of dimension $4k$ then the $\hat{A}$-genus vanishes. By calculations with the spin bordism group, there is a sense in which a "generic spin manifold" of dimension $4k$ has nonvanishing $\hat{A}$-genus (at least for $k > 1$ - not sure if anything weird happens in dimension $4$), so the conclusion is that a generic compact spin manifold of dimension $4k$ has only discrete symmetries.
By contrast, while many (but not all) of the basic examples of compact manifolds one encounters in everyday life are spin, most have some sort of continuous symmetry (indeed, quite a few are homogeneous spaces).