Yes, they can deplete. More modern games (starting with the Nintendo 64, for instance) used EEPROM (similar to flash memory) so don't have this issue, but in the days of the SNES this was too expensive so battery-backed SRAM was used.
How long the batteries last depends on the game. Pokemon Silver for example has a real-time clock, which takes a lot more power than SRAM alone, so all copies of it you'll find will either have already run out, or had the battery replaced at some point. On the other hand, games that just have SRAM might still be going strong today. But if your save game is important to you you definitely shouldn't rely on the battery lasting. Use one of the many methods available to back it up.
After making a backup, the battery can then be safely replaced without risking data loss. Most of these batteries are soldered in place, unfortunately, so it's not always easy to change them; you can often solder in a battery holder to make future replacement easier, if there's enough space in the cartridge. Most of them are CR2032 or the very similar (and often interchangeable) CR2025.