One problem related to transmission pump has been reported for the 1996 Chevrolet Cavalier. The most recently reported issues are listed below.
At 46k miles, my transmission panel light started going on and off. At 53k miles, a gm dealer read a p1887 tcc release switch circuit malfunction. At 58k miles, the transmission broke at 70 mph on i77 north of charleston wva. The gm dealership in wva found that the pump in the transmission came apart, damaging the clutch and flywheel assembly, replaced the transmission, and billed me. I called gm customer service, they said I was out of warranty, tough luck, no other problems like this have ever been reported. However, I found numerous identical problems in the NHTSA files (see NHTSA bulletin sb049140). I printed a TSB (service bulletin) from a gm dealer in ohio that shows that the transmission pumps installed on Cavaliers before 6/96 have had transmission pump seal failures that can lead to the transmission burning up even though the car drives like normal. Since gm admitted in the TSB that cars produced before 6/96 (mine was built 4/96) had a pump problem, and there is no recall on the Cavalier transmission, I believe I still should be reimbursed by general motors for the cost of a replacement transmission on my car. The cost from the dealer to replace the transmission was $2,062. 70.