What Bernie Sanders' String of Victories Means for Hillary Clinton in New York

The Democrats are facing off in a state where they both have ties.

"The question is if this is putting pressure on the superdelegates to rethink their support for Clinton," he added.

After his latest victory in Wyoming over the weekend, Sanders said on "This Week" that "there's no question I think the momentum is with us."

Phillip Wallach, a senior fellow in governance studies at the Brookings Institution, said that it does "certainly feel like Clinton is on her heels," but he doesn't think it will last.

Moving forward, it is "obviously very important to her to win a majority of pledged delegates," Wallach said.

"To win on the strength of the superdelegates would cause some big headaches for her," he added.

The next contest is in New York, a state where both candidates have ties, on April 19, and it could have major implications for both campaigns even if Clinton does win. The latest state poll, released by Quinnipiac University on March 31, had Clinton leading by 12 points.

"If she were to lose, or if it were very close in New York -- a state that she represented in the Senate -- that would be very devastating," Campbell said.

Campbell noted that even if Clinton won but Sanders still had "a respectable showing" of upwards of 40 percent of the vote, that could lead to problems for the Clinton camp.

"She might be able to survive that, and she probably would be able to survive that, but it would make it more difficult and it might have superdelegates rethinking their commitment to vote for Clinton on the first ballot," Campbell said.