Courant's nodal domain theorem gives a bound on the number of nodal domains for an eigenfunction of the Laplacian. Namely, if $M$ is a smooth compact Riemannian manifold, and $f$ is an eigenfunction for the $n$th eigenvalue, then the number of nodal domains is bounded by $n$.

Is there a bound on the number of nodal domains for a sum of eigenfunctions (with different eigenvalues)?

Note that in the case $M$ is a sphere, any linear combination of eigenfunctions (spherical harmonics) is a restriction of a homogeneous polynomial of degree $d$ (which is roughly the square root of the largest eigenvalue) and Harnack's bound says that there is a bound of order $d^2$ (same order as Courant's bound for the top eigenfunction). Similarly, higher dimensional spheres and flat tori admit bounds with the same order as Courant's that apply for a sum of eigenfunctions.