Usually if it were a bad battery saving would not work at all, I think. On the back of all/most cartridges with battery saves there was a warning to hold down the reset button when turning off the Nintendo. Somehow when you would press the power button to turn off the system there was a possibility of blowing out the memory on the cartridge. I had heard of a lot of people who had experiences that problem, but I never bothered with holding down the reset button and I never had it happen to me even once. Maybe some cartridges were more likely to experience it than others. That could have been what was happening with you and your brother with Baseball Stars.

Kind of stupid Metroid, Contra, Metal Gear, Jackal and the later Mega Man games had passwords, while sports games got to have battery saves. But I guess password saves wouldn't really work too well with sports games. The Japanese Famicom version of Metroid had save states though.