Yes that is totally the point, but the compressor might have a high and low run time setting. Some compressors will come on, and stay on for a certain time period and when they shut off will stay off for a time period regardless of the call for cooling. Its a feature to help keep the compressor from burning up from on--off-on-off.
This may not help, but try widening your setpoint tolerance to 10-15 degrees, it may allow time for the compressor to reset. If it's old enough, you may even be able to find the timer and remove it. Use Google.