I own an old house that I am currently renting out (I know live in another state). The renter called me and said that the radiator in a upstairs room (it's a two-story house) has not been working. I sent my brother-in-law over and he added some water to the system (until the overflow valve activated) and then the radiator work - to a degree. NOW the radiator heats, but only on one side (the side where the inflow pipe is) and is cold on the side where the outflow pipe is). Now this seems strange to me - there can't be a blockage in the radiator, otherwise no hot water at all would get into that radiator? Or at least the radiator would heat on the BOTTOM half and not the top half, rather than left and right?

Do I just need to have more water added to the system? Perhaps have someone bleed the upstairs radiator while more water is added to the system?

Or is there some dramatic issue that I'm not seeing?

Usually air in the system or low system pressure is the cause. Now for a two story usually 12 to 15 pounds does it as far a water pressure/altitude is concerned.
The pop off safety is usually preset to 30 psi so he added water to bring the perssure over that amount and that is a no no..
Also a poor performing pump
A bad auto fill valve
A water logged expansion tank and other problems can cause this type of a problem.