Yes. That's how they do research on Alzheimer's. An line of rats have been engineered to express Alzheimer's symptoms. They are literally born with it. They are the control use to test Alzheimer's treatments.
To be technical, they may not have Alzheimer's. They were engineered to have the plaques that are associated with Alzheimer's. They do show Alzheimer's symptoms just like in humans. But whether those plaques cause Alzheimer's is an open question. No one knows what causes Alzheimer's.
Here is a link to PBS special on early onset Alzheimer’s in families. It might be helpful.
http://www.pbs.org/program/forgetting/
My boss, an RN, explained it this way.
Dementia - You know its a stove but you cannot remember how to work it
ALZ - you don't remember its a stove.
To be technical, they may not have Alzheimer's. They were engineered to have the plaques that are associated with Alzheimer's. They do show Alzheimer's symptoms just like in humans. But whether those plaques cause Alzheimer's is an open question. No one knows what causes Alzheimer's.