For over two years, I kept my day job despite earning enough for a full-time income through affiliate marketing. It was a great job as the marketing director for a fun e-commerce company, and I liked the variety. Plus, it was cool to view things from the perspective of a retailer, learn more about direct mail, and get experience in things I wouldn't have tried otherwise.
Recently, I started to notice that I was always tired, I never felt like I had enough time for anything, and I wasn't learning all that much at the day job. I gave it a few weeks to make sure it wasn't just a passing mood, and finally, I just quit.
It wasn't really a calculation of a certain amount of money in the bank, for two reasons. First, I was making more on the side than at the day job. By quite a bit. Second, I have always picked up side jobs in writing/consulting/etc., so I knew I could always do a little more of that if I really needed to at some point.
If you're genuinely determined to make your living online, I would strongly recommend learning some kind of secondary skill that you can use to earn a bit of extra cash if you really need it. That way, even if you find yourself short on money to reinvest in PPC for a few months, you can sell increments of your time to make up the difference. In a lot of cases, it's much easier to do this than to save up some massive stockpile of cash.
Now that I've quit, my only regret is that I didn't do it sooner. Even if you enjoy your day job, you're leaving a lot of money on the table by not devoting yourself to your affiliate efforts as soon as you can reasonably do so. Of course, if you don't have a second marketable skill, you'll probably want at least 6-12 months of living expenses plus a few thousand to start your PPC efforts.