Hi, my condolences on the loss of Tyler. It is never easy to lose a loved one.
Pregnisone is not a familiar name to me. Do you mean prednisone?
Although it is contra-indicated to use prednisone at the same time as Rimadyl, it doesn't sound like the problem from doing so would be paralyzed back legs, it would be stomach irritation or ulceration, altered kidney function, or inappropriate bleeding. That suggests the throwing up could have been from the two drugs interacting, but it seems less likely the paralysis was. How old was Tyler? What kind of dog? What kind of health history?
Did you discuss why Tyler's problem might have occurred with the Texas A&M people? They would be best qualified to tell you if there was even the remotest chance that the two drugs in combination could produce such a dire result as paralyzed back legs.
Again, I'm sorry about the loss of your beloved pet.