Do exhaust manifolds go bad

Absolutely, exhaust manifolds can go bad. The failure is usually not catastrophic though, meaning a car will remain driveable. Failures typically occur as a crack in the seams between the individual headers that are caused by cyclical heating and cooling.

The result is usually a noticeably loud hissing sound, fouled oxygen sensors or a decrease in fuel economy to name a few things. Consult manufacturer reports for cars that may have had recalls.