Why might an altimeter read incorrectly by 4000ft after sitting on the ground for some time?
This is how it works and I am surprised you switched the altimeter, unless I am missing something in your question. The altimeter reads the barometric pressure and should be set before you go flying. The airport will report the elevation and the knob on the altimeter can be turned till the altimeter is set to the correct altitude. You can confirm there is no blockage and the altimeter works correctly by getting the barometric pressure for the airport from the control tower or weather station, and it should match the barometric pressure numbers in the corner of the altimeter. Pressure changes on a daily basis and must be set each time. When we do a B check before flight, this is the exact procedure we do. 

There is more information needed to properly diagnose this problem. With the lack of information I will continue on a couple of assumptions. 

1) The altimeter has been bench tested and is serviceable
2) The altimeter system when installed in the plane does not match the local barometric pressure with the corresponding elevation

It was stated that the altimeter was replaced with a known good one. When this occurs the whole system must be pressure checked with a pitot-static test kit. This is a requirement since the system was opened up by replacing the altimeter. If this is not done the system is considered unserviceable. It appears this was not done.

To be complete I would check out the lowest point in the system for moisture and drain it if present.

Generally the procedures including connecting the tester to the pitot static system and taping (covering) all ports not connected. Then pressure is applied to the system, simulating different elevations. The airplane altimeter should match the testers simulated elevation within a certain + or -, for all specified elevations. Additionally, the rate of change of elevation is monitored for any abnormalities that may indicate a problem with any internal mechanisms.

Using the pitot-static tester is an absolute requirement in this situation.