I previously posted an issue after install of a Nest Learning Thermostat (3rd Gen) where even though in Heat mode only, my fan was blowing at random intervals. My HVAC system is separated (heat is radiators not forced air) and the fan is part of the CAC system. The problems I am having with my HVAC system, specifically, the fan blowing, did not occur prior to installation of the Nest Learning Thermostat. When I contacted Customer Care, I was advised that the installation of a C wire may resolve the issue. I specifically purchased the Nest Learning model because I knew I did not have a C wire in my current wiring. Customer Care advised they would provide a promo code for a free install of a C wire via OnTech to resolve the issue. Initially, OnTech came to my home and did not believe installing a C wire would resolve the issue. I had a HVAC tech, at my own expense, come to my home and advised a C wire probably would resolve the issue because the thermostat was pulling power cause the fan to start up. I again scheduled a C wire install via OnTech using the promo code. The tech came out and advised because the wiring was inside the walls, they could not provide the install. It is important to note that the C wire install by OnTech normally costs $129. After I advised that OnTech would not install the C wire, Customer Care advised another solution would be purchasing a Nest Power Connector and having it installed at my expense. Notably, OnTech also charges $129 for the install.
My point is that I am past a return window for the thermostat, which is causing a problem I did not have before. Customer Care was willing to absorb the $129 C wire install (which was not done) cost but now Customer Care unwilling to absorb the Power Connector install cost which is the same? I am willing to purchase the Nest Power Connector at my own expense even though this was not advertised as needed for this model. I feel it is Google/Nest's responsibility to resolve the problem with the product.
Why dont you just pull a new 5 conductor thermostat cable from the AC system to the Nest and connect a real C wire?
You just tie the new cable onto the old wire really securely and slowly pull it up into the attic I assume where the ac air handler is located.
Then hook up blue to C and match up the rest as they are now.
You can buy the cable at Lowes and Home Depot etc comes in 25ft or 50ft roll in a bag.
Get 5 conductor.
This will be a better solution in the long run because all new systems need a C wire so if you ever replace the system it needs 5 conductors.
Where did you install the power adapter in the AC system or the boiler?
The problem is due to the fact that you have a dual transformer system.
The AC has a transformer and the boiler has a transformer. You have no C common back to either transformer to complete the electrical circuit so it tries to bleed current off the other wires to maintain the power to keep the Nest charged.
The solution is to pull a new wire back to the AC system and connect it to the C common on the terminal screw and the Nest.
Your old wire is probably loose inside the wall and you can tie a new thermostat cable to the old one and slowly pull it up thru the wall into the attic I assume where the AC unit is located.
If the Nest is on 3rd floor they probably drilled a hole and dropped the thermostat cable down from the attic to the thermostat.
It should be easy to replace the AC cable going to the attic. Just get a 50 ft roll of 5 conductor thermostat cable at Home depot. Securely attach to the old cable and pull it down from the attic or up thru the wall into the attic, Then attach blue as a real C wire in the AC unit.
If you insist on using the power connector I would put it in the AC system because the boiler probably has no C terminal screw and you need a C terminal for the power connector. The time and effort you will spend hooking up and rewiring for the power adapter you could have ran a new 5 conductor cable and been done with the problem!