Recent stock market gains could be on the verge of vanishing, in part because "June is a very funny month," according to UBS' Art Cashin.

"We talk about sell in May. That didn't work in May to begin with. June is a difficult month to get real big gains," Cashin said Tuesday on CNBC's "Trading Nation."

The math shows June is indeed the worst month for stocks.

According to the latest data from Bespoke Investment Group, June has been a down month for the Dow 60 percent of the time over the past 20 years, with the negative returns ramping up in 2001.

During the past half century, the Dow has fallen in June more than half of the time, according to Bespoke.

Cashin, UBS' head of floor operations and a fixture at the New York Stock Exchange since 1964, says he's detecting ominous activity that could indicate pain ahead. He says rising and bond yields falling is concerning.

"That's a real dichotomy, particularly with a pretty heavily discounted Federal Reserve hike coming up in about a week," Cashin said. Fed policymakers meet next Tuesday and Wednesday.

Cashin said bond yields could be moving down due to signs that the economy isn't as robust as anticipated, and he pointed to mounting jitters that economic growth could fall back below 2 percent.

A scenario like that could alter the Federal Reserve's current interest rate policy, and Wall Street doesn't like surprises.

"There are some people that think that yield on the 10-year [bond] could slip below the 2 percent level again, and that would certainly get everyone's attention," he said.