Some museums are more comprehensive than others. Nevertheless, some specialization is necessary due to inevitable budgetary limitations. And then there is the factor of donations of art over the years.
Other specialization is voluntary. For example, New York's Museum of Modern Art, as its name states, focuses on modernism in its various guises. And the Albright-Knox Art Gallery in Buffalo, New York set in motion a de-acquisition program a few years ago in order to focus on art of the recent past (scroll down in the link for more information).
I suspect most art museums fall into the former category with some added policy steering by whoever is running things at any given time. The result of all this is a museum's character as perceived by visitors. For instance, the largest art museum in my neck of the woods is the Seattle Art Museum. The downtown building has some paintings from 1500-1700, a lot of modernist art and a decent collection of art by Northwest "mystic" painters such as Morris Graves and Mark Tobey. What is almost completely missing is 19th and very early 20th century art, though some bequests might improve matters eventually. The Portland Art Museum down the freeway in Oregon is much stronger for that period.
Another west coast museum in an even smaller metropolitan area that has some nice late 19th and early 20th century paintings is the Santa Barbara Museum of Art, described in the link. Its web site is here, a link to its European collection is here and the American collection link is here.
The collection links are worth exploring because they contain thumbnail images of many (most? all?) the works in each collection; I saw a number of these recently while visiting Santa Barbara. To me the most impressive work that I don't illustrate below is William Merritt Chase's portrait of his wife, a large painting that might or might not be finished -- detailing other than the face is somewhat sketchy.
Here is a sampling of items in the collection. Aside from the Monet, none are well-known, but they provide viewers a decent idea as to what some famous artists were doing.
Gallery
