Maybe if it was just Nintendo and Sega, they might have but it was more a matter of money than anything else. Even the Gamecube didn't have much third party support compared to the competition. Even worse than that was the N64. But they were well off financially and they pulled through.
If not for Sony buying exclusives, remember that MS was still there. Then again, if it wasn't for the Playstation, we don't know if there would have been XBox.