Most often it tends to be the more prestigious institutions that grok the possibilities first – the Smithsonian, the British Library, etc. – and which engage wikipedians-in-residence, run edit-a-thons, donate images & video; the GLAM site has some case studies.

Apart from the obvious commonality of mission, there’s some evidence (citation needed) that engaging with wikipedia can drive click-through to institution websites. Generally it’s a win-win proposition.