We had a recent situation where we located a cased image that had been stolen from our museum many years ago. It had been purchased by a collector who sits on the board of a museum in another city with a similar theme and mission as ours. He had to return the photograph to us. There is also a house museum here that has had incidents of family objects associated with the house coming up for auction, and board members who are also collectors have outbid the house on purchasing these objects!
I think board members who are collectors is not such a great idea.