I know on most cars, a dummy light will come on when the voltage drops below a certain point. It doesn't normally point to a specific component, but the charging system in general. I would think that the scenario you describe is possible.

'88 Raider. 2.6L dead...2.4L swap in progress, AT, 33's, 2 bouncy seat,AC, CC, 2.25 exh, 2" Tbar, 2" BL, rear DB, rear LSD,4.90's, SW hubs,gen2 uca's, KYB shocks, snorkel, elec fan, MSD.

'99 chev k3500 crewcab
'15 Subaru XV
'03 BMW i325