A couple of years ago, my wife & I visited the Lowry Gallery at Salford Quays. Entry is free, but as you arrived at the gallery you had to visit a ticket desk which had a prominent notice giving a "suggested" contribution (may have been £5 IIRC). The desk attendant asked how many visitors & handed over the appropriate number of entry tickets, which were elaborately printed like a pricey concert ticket, possibly even with a metallic security strip.
The whole impression given to arriving visitors was that there was an entry fee, although of course nothing stated you had to pay.
I was wise to this trick from researching the Lowry online prior to going (a reviewer had commented unfavourably on the mild deception), and the desk attendant seemed a bit surprised when I just took the tickets, said "thanks, pal" and walked in without paying.
We both enjoyed our visit, so much so that on the way out I dropped money in the box to the value suggested.
I certainly don't mind a request for a voluntary donation at my discretion, but I didn't like the strategy of implying to mug punters that there was an entry fee when there wasn't.
Looking at the Lowry website today, they make it perfectly clear that entry is free, and maybe they now make this a bit clearer at the entry point, I don't know.
Are there any other free museums that do this? What's the situation at the NRM in York?