I have a 1996 Honda Civic LX with an electrical problem?

Several weeks ago, my battery light came on. It goes on and comes off and goes back on all the time. The alternator and battery have all been checked and are working properly. The connections are good as well. Recently, the battery light came and stayed on, the radio went haywire (eventually turning off), the air conditioner went off, the lights on the dash went out, the speedometer stopped working and the car started to hesitate. There is 189,000 miles on the car and it is a 1996. What could be the problem?