My 1987 Honda civic lights would go dim and so would the dash board. We had changed the alternator and checked the battery what could this be?

. Remove and fully charge the battery. Most auto parts stores will do this for free. Many "battery tests" are superficial and not reliable. Instead, remove the battery, check electrolyte levels, place on a charger, and watch the ammeter. With a good battery, the ammeter will soon cut-back to 1 to 2 amps.

. Check that the alternator's regulated voltage is 14 to 15 volts, while idling. If it's low, install another alternator.

My bet is that there's a problem with the battery.