This isn't a right way, wrong way question just a what works for you question.
I personally, love art, but can't absorb it for more than two hours (three at the outside) at a time. So I try not to see more than one art museum a day and to see it for about two and a half hours. Not long ago on a college looking blitz my daughter and I had a three, three and a half hour chunks of time in Chicago spread out over three days and we spent them all at the Art Institute. It was one of my best museum experiences ever. Each day we spent two hours museuming, and an hour and a half eating lunch and looking at the gift shop. It was perfect. I enjoyed what I saw much more than I would have if I'd spent a whole day there.
I love museums like the D'Orsay, the Uffizi, or better yet the Rodin, that are just naturally two to three hour visits. Faced with the Louvre, The National Gallery, The Kunst Historiches, my brain locks. My solution is to spend two or three hours there in specific wings and plan on coming back later either on that trip or assuming I'll be in the city again.
I might alternate an art museum with a history museum, but I usually regret it if I do history or art museums back to back. Half a day in museums is about right for me, but if I see a whole day of museums it had better be half art and half something else. But a half day only of museums of any kind is best. After that, I'm better off just walking, or climbing towers, or shopping, or taking a bus or boat tour. Those are memories I'll take away. Five or more hours of museums are wasted on me.
How do you handle museums and how much do you want to see at once? Whatever the answer is, it's worth thinking about when planning your days.
What works for you?