One can claim tax exempt status and not have any withholding on paychecks.
The W-4 form reads:
"I claim exemption from withholding for 2009, and I certify that I meet both of the following conditions for exemption.
Last year I had a right to a refund of all federal income tax withheld because I had no tax liability and
This year I expect a refund of all federal income tax withheld because I expect to have no tax liability.
If you meet both conditions, write Exempt here"
I used this option in graduate school before (when I was making less than $25K a year) and it was a royal mess. The IRS wants quarterly reports to assure you're actually exempt, and if you owe any tax at the end of the year you'll pay late fees on that amount. I'm not sure the tax on the signing bonus is related to the number of exemptions you claim, but I wouldn't claim tax exempt status unless you're really really sure.