We study conditions for the containment of a given space X of analytic functions on the unit disk D in the de Branges-Rovnyak space 9L(b). We deal with the nonextreme case in which b admits a Pythagorean mate a, and derive a multiplier boundedness criterion on the function 0 = b/a which implies the containment X C 9L(b). With our criterion, we are able to characterize the containment of the Hardy space 9Lp inside 9L(b) for p E [2, oo]. The end-point cases have previously been considered by Sarason, and we show that in his result, stating that 0 E 9L2 is equivalent to 9L infinity C 9L(b), one can in fact replace 9L infinity by BMOA. We establish various other containment results, and study in particular the case of the Dirichlet space D, whose containment is characterized by a Carleson measure condition. In this context, we show that matters are not as simple as in the case of the Hardy spaces, and we carefully work out an example.