Although the creation of spin polarization in various nonmagnetic media via electrical spin injection from a ferromagnetic tunnel contact has been demonstrated, much of the basic behavior is heavily debated. It is reported here that, for semiconductor/Al2O3/ferromagnet tunnel structures based on Si or GaAs, local magnetostatic fields arising from interface roughness dramatically alter and even dominate the accumulation and dynamics of spins in the semiconductor. Spin precession in inhomogeneous magnetic fields is shown to reduce the spin accumulation up to tenfold, and causes it to be inhomogeneous and noncollinear with the injector magnetization. The inverted Hanle effect serves as the experimental signature. This interaction needs to be taken into account in the analysis of experimental data, particularly in extracting the spin lifetime $\tau${}s and its variation with different parameters (temperature, doping concentration). It produces a broadening of the standard Hanle curve and thereby an apparent reduction of $\tau${}s. For heavily doped n-type Si at room temperature it is shown that $\tau${}s is larger than previously determined, and a new lower bound of 0.29 ns is obtained. The results are expected to be general and to occur for spins near a magnetic interface not only in semiconductors but also in metals and organic and carbon-based materials including graphene, and in various spintronic device structures.