It takes roughly 1 hour and 45 minutes to replace the motherboard in a Dell Latitude D820.
A couple weeks ago I called up Dell because I noticed that sometimes when I picked up my laptop, it would turn off. Dell figured the appropriate course of action would be to RMA the motherboard and power supply. So tonight I held my breath and did it (I prefer to do these things myself).
In retrospect, my battery was probably just loose. Go figure.