First, what's the reason? Why do you have that much in cash at all - are you concerned about market volatility, are you planning to buy a house, do you have tens of millions of dollars and this is your slush fund? Are you a house flipper and this is part of business for you?
If you need the money for short term use - ie, you're buying a house in cash next month - then as long as you're in a sound bank (one of the big national ones, for example) it seems reasonable. You can never predict a crash like 2008, but it seems unlikely that Chase or Citibank will go under in the next few weeks.
If you like to have a cash position, then split the money among multiple banks. Buy a CD at one major bank with some of the amount. My in-laws have a trust which is partially invested in CDs, and they use multiple banks for this purpose to keep their accounts fully insured. Each separate bank you're covered up to 250k, so if you have $150k at Chase and $150k at a local bank, you're covered. (You're also covered in a much larger amount - up to 1MM potentially - if you are married, as you can have a separate account each for $250k and a joint account up to $500k.)
Otherwise, why do you have that much in cash? You should invest it in something that will return more than inflation, at a minimum...
Edit post-clarifications: $350k is around my level of 'Maybe, maybe not'. You're risking $100k on a pretty low risk (assuming this isn't a small local bank, and even those are pretty low still). In order to remove that risk you have to do something active - ie, take 100k somewhere else, open a new bank account, etc. - which isn't exactly the hardest thing in the world, but it does take effort. Is it worth the 0.001% chance (entirely made up) you lose the 100k? That's $10, if you agree with that risk chance. Up to you.
It wouldn't be particularly hard, though, to open an account with an online bank, deposit $100k in there in a 6 month CD, then pay the IRS from your other account and when the 6 month CD expires take the cash back into your active account. Assuming you're not planning on buying a house in the next six months this should be fine, I'd think (and even then you'd still have $150k for the downpayment up front, which is enough to buy a $750k house w/o PMI).
Additionally, as several commenters note: if you can reasonably do so, and your money won't be making significant interest, you might choose to pay your taxes now rather than later. This removes the risk entirely; the likely small interest you earn over 3 months may be similar to the amount you'd spend (mostly of your time, plus possibly actual expenses) moving it to another bank. If you're making 2% or 3% this may not be true, but if you're in a 0.25% account like my accounts are, $100k * 0.25% * 0.25 is $62.50, after all.