I think the basis of this rule is to have similar behavior of all bearings. For instance if one was to combine pot bearings and elastometic bearings on one substructure unit, the pot bearing would be stiffer for vertical load. This could cause stress concentrations and other problems. If there is a compelling reasons, I think you can combine bearing types, but you have to use good judgment with the details to avoid problems with different stiffnesses.
What do you mean by 'type'? say pot v.s. elastomeric? Or fixed v.s. sliding? or manufacturer?
I have worked (not designed) on a number of bridges, most of which had several types of bearings. In fact to have fixed bearings on one pier or abutment and sliding bearings on the rest seems to me a reasonable arrangement.
In the project I am currently working on I have pot fixed, pot sliding, elastomeric, knuckle bearing, a strange mechanical german bearing... Mind you, it is quite a complex bridge.
I have not heard about the rule you mention, unless you are talking about the bearings on the same pier or support, which would make sense to me to keep the same.
All the same, Graybeach is correct, whatever bearing arrangement is chosen, I assume that it has to be reviewed and analyzed to ensure it is working as a system.
