If you do it wrong, yes, you can kill a bush by killing it.
In general, you should avoid trimming or pruning a bush in the fall time. The freshly cut ends may die as the temperature drops. It will also not grow back, which may have been your intention when you pruned the bush. This is because plants don’t use the same energy in the fall.
A bush can also die if you make a flush cut. A flush cut is when you take off a branch flush with the bark or with a larger branch attached (source). This takes away the branch collar that provides essential tissue for a seal to form over the pruning cut. Now, the plant cannot heal and you have a flush cut. This is an ideal place for bugs and other diseases to enter and kill the plant.