Can two photons excite an electron consecutively?

To add to the answer by @CrazyGoblin. Every state can be characterized by a lifetime, $tau_i$, which characterizes the rate of relaxation $Gamma_{irightarrow 0}=1/tau_i$ to the ground state. Inducing an absorption to a higher energy state requires that the absorption happens faster than the relaxation. In other words, the rate of absorption, given by the Fermi Golden rule, $Gamma_{irightarrow j}$, should be higher than the rate of relaxation to the ground state (preferably much higher, for the effect to be clearly observable): $$ Gamma_{irightarrow j} gg Gamma_{irightarrow 0}. $$ Since the absorption rate is proportional to the square of the matrix element, i.e., to the square of the field inducing the absorption, this usually requires very strong optical fields, and as such belongs to the domain of nonlinear optics. The absorption from the ground state obviously has the advantage that the ground state has infinite lifetime, and the absorption can be easily observed even for relatively low fields.

Finally, let me note that given strong fields and appropriate non-dipolar couplings, one can observed two- and multi-photon absorption even with no intermediate levels present.