Basically, the problem is with the way many museums label their exhibits. Frequently there is a number next to the object, and this refers to a panel with explanatory text. This is OK for small objects where there is no room to add captions next to the objects, but frequently it is employed for large objects, where the aesthetic value of the object is often seen as more important than its meaning.
Unfortunately this makes it very difficult for people with dyslexia to enjoy the exhibition, because by the time they have transferred their gaze from the exhibit to the interpretation panel, they have forgotten the number, and have to go back again. I myself am not dyslexic but frequently have this problem anyway! In one museum we visited, there was a costume exhibition, and the distance between the numbers and the explanatory text was so great that a lady who was both short- and long-sighted had to change her glasses each time.
The solution is simply to place a short explanatory caption next to the object (e.g. 14th C English spoon), and a longer piece giving the context below or beside the display case.
A quick Google search reveals that many museums are concerned about web accessibility and physical access for wheelchair users (and rightly so) but many museums appear to have completely overlooked this problem of captioning.
Obviously, objects must have an accession number, but the problem of captions for visitors is different.
An additional problem is that different audiences want different information about objects. Personally, I find the social context of objects interesting. Others might be looking at them from an art history perspective, or a comparative culture perspective, or some other perspective.
Here's an example of a display case that is accessible for dyslexia (it's from the Pitt-Rivers Museum in Oxford). Note how the labels and descriptions are next to the objects (no cross-referencing required):
How not to do it (this is an imaginary display that I have constructed of objects from the Science Museum):