For some reason the circuit breaker is tripping. This has nothing to do with the Nest thermostat. The thermostat is letting you know there is a loss of 24 v power. It has a built battery in the the event of a power outage. It has a diagnostic software to alert you to problems like this.

You need to figure out why the circuit breaker is acting up. What type of heating system do you have?

In my system, without the common ground, the thermostat lost connectivity to the Heat Pump, and would reset with a 5 minute delay. I was thinking that loss of connection, could cause the trip in their system.

What Mike said is correct and to help you see it clearer.
The 24 volts are supplied from a transformer which is dependent on same 240 or 120 volt that powers equipment. Each equipment device must be amp checked to find what is drawing excessive power and triping breaker. 1 time out of a thousand the breaker can develop a problem that requires a new breaker.

Thank you so much for the responses so far. It is definitely not the thermostat. Just when the weather turned cold this year we lost power to our old thermostat and called our friend who works in HVAC to come take a look. He tripped the breaker which fixed the issue and that is how we knew to do that this time around. When researching causes my husband decided he wanted to Nest and we got it mostly for aesthetic purposes. So, definitely not the Nest or its set up causing the issue.

Bare with me as I am new to all of this but will try my best to describe our system. I thought it was Carrier but was wrong. It is a 1988 Amana all-electric heat and cooling system. It was a tested factory model that the previous owner got from her father who worked at the plant and it was installed in September 2001. The breaker is in the crawl space and we have a 1700 square foot two story home. Are there any other details that might be helpful?

My husband has a good friend who has worked in HVAC repair for several years. He is a friend but still not cheap! I think we may just call him back over to test what is pulling so much power that it is tripping the breaker, but thought I would check here first for some tips. Thank you everyone!

I assume this is a heat pump and not an electric furnace.

The equipment is over 25 years old. I don't know what factory tested means. Was it under test for 13 years at the factory? Whatever the case it is time to plan for a replacement.

When your HVAC friend reset (not tripped) the breaker, did he make any comment as to why this may be happening?

Thanks, Mike!

Yes, it is a heat pump not an electric furnace.

I am not sure of the circumstances really. But 13 years or 25 years, either way the thing is at the end of it's expected life.

Yes, he said there was some sort of power surge and asked if we had seen a jump in our electric bill. We haven't seen any spikes really, but this is only our second winter in the home. He came at 6:00 pm and it was already dark (and super cold) so there was not too much he could really look at.

I am thinking we will have him come back during the day to look at the unit and test a few things, and probably get a quote for a whole new system. Hopefully we can repair our current system and not have to shell out too much.

many furnace guys will not even look here but try watching this guy and then ask the HVAC guy to do this if you dont want to. https://youtu.be/Nl_FAGqhQ9I

Works like a charm, i replaced my entire furnace after having a similar problem, the problem seems to go away but thats because it was the summer and heating was not on. Watch this and share!

Old thread, but sounds like something in the heat pump has gone bad and is drawing too many amps, either blower motor or condenser pump. Need someone to diagnose with a clamping ampmeter. http://www.homedepot.com/p/Klein-Tools-400-Amp-AC-Digital-Clamp-Meter-CL1000/202266605