(Read more: Twitter IPO 'could be bubble in a heartbeat': Dan Nathan)
"Today is not a day to throw everything you've got at it. I think you'll have other opportunities," he added.
On CNBC's "Halftime Report," Brown said that Twitter's IPO was notable given the broader market's weak performance.
"This incredible open for Twitter is happening against a backdrop in which all 10 S&P sectors are red on the day, and the tech sector is actually the worst performing, down 1.1 percent," he said. "So, this is a good day for capital formation. It's great that a company like Twitter can come public, get this kind of reception, even in the midst of a little bit of gloom."
OptionMonster's Jon Najarian said that he had received an allocation of Twitter stock at the IPO and was looking to buy in the $46 range to flip around $48 per share.
(Read more: Twitter more like Pandora than Facebook: Pro)
"If you're a trader, this is manna from heaven because this is a trading stock," he said, noting that in morning trading, Twitter stock was seeing 2,200 trades per second.
Rosecliff Capital's Mike Murphy said that it appeared Twitter stock was finding support at $45.
"I think it's interesting if it cracks that, where the next level of support is going to be," he said. "We don't know that yet."
Asked if Twitter was a stock that retail investors could buy, Murphy said, "Absolutely not."
TheStreet CIO Stephanie Link said that there was "some value" in the long term.
(Watch video: Buying Twitter like a pro)
"Short term, this stock is now trading at 22.3 times price to revenues. That's on next year's revenues," she said. "Maybe they might beat those revenues, but you're looking at Facebook trading at 11 times price to revenues, and we have a little better visibility on how they're going to get to those revenues."
Twitter, she added, would have to reverse its trend of decelerating user growth to see a higher share price.
"There are some question marks," she said. "Great growth story. I just wouldn't buy it up here."
—By CNBC's Bruno J. Navarro. Follow him on Twitter @Bruno_J_Navarro.
Trader disclosure: On Nov. 7, 2013, the following stocks and commodities mentioned or intended to be mentioned on CNBC's "Fast Money" were owned by the "Fast Money" traders: Stephanie Link is long AAPL, JPM, MS, CSCO, FB; Jon Najarian is long AAPL, GLD, MSFT, TSLA, IBM, FB, TDC; Josh Brown is long AAPL, F, TWTR; Michael Murphy is long AAPL, BAC, C, FB.