the answer is "it depends", but it's true that Weaver will usually fit on a Picatinny rail, particularly for rings. Mounts are trickier.
Weaver rail is a design, not a spec. Slot and crossbar dimensions and spacing vary a lot, even within Weaver's designs. In general, the slots and crossbars are narrower than the Picatinny spec, and the spacing is totally variable. Weaver bars fit in 1913 slots (loosely), but not all 1913 compatible bars fit in Weaver slots. Many manufacturers purposely make the crossbars small enough to work on either slot, like on the Eotech holosights.
Picatinny (real name - MIL-STD-1913) is a spec based on the Weaver design, with tolerances. Any true Picatinny spec gear will work with other true Picatinny spec gear, but not all makers hold to the spec, just like with Weaver. Most often, the spacing and/or slot width are different for out of spec gear. Even ARMS, who some credit with developing the Weaver design into the Picatinny style design that was turned into the MIL-STD-1913, makes non-Picatinny spec mounts.
The cases where Weaver style gear won't mount on a Picatinny spec mount usually involve gear with 2 fixed position crossbars. Since the spacing is not controlled in the Weaver design, you can have a case where one crossbar fits in a slot and the other doesn't.