OP, you did not provide enough information for people to give you appropriate advice. E.g., people are planning on worst case scenarios - that you lose your job, or otherwise are unable to work. The likelihood of that varies tremendously from person to person. Some jobs have a lot of job security. Okay, what about getting sick - my wife and I can each be sick for at least 6 months without any loss in income. Hers is a little less accrued sick time, as she's allowed to cash in half every time she hits 500 hours; but she has better long term disability that would be supplemented by sick pay.
Like any investing, there are varying degrees of risk. Those who say "don't use any of your cash reserves" apparently believe in absolutely zero risk. But, if you're like the average person, using all your cash reserves does put you at increased risk of being screwed for years - regaining the debt, plus when you're late after the first 30 days because you can't make a payment, you get late fees + the default high interest rate.
But, perhaps there's a happy medium somewhere in between... Pay a significant chunk of those cards, while leaving enough cash to cover you for a couple of months. How long you need to cover yourself depends a bit on the type of job you'd be seeking in case of something like a layoff. If you presently work at McD's, screw it - pay it all off and consider that you'll get hired at BK or Wendy's the next week. But, if it would be expected that it might take several months to land a new job, then you should consider reserving more of that cash. Still, though, after paying down a significant chunk, you'll have lower payments each month, so can afford to rebuild your savings quicker. Finding the right happy medium is up to your individual circumstances. The more risk you take, the higher the payout could be for you. The less risk you take, the more you end up spending on interest. How much risk you should take depends on your age, health, type of career, monthly expenses, how well your company is doing, etc.
edit: and if you're plunking down a chunk of your savings, but not paying them all off, pay off the higher interest rate ones first. Don't just divide up the money evenly across the cards.