Weston Super Mare are two divisions and 41 places higher than Town, but are currently managerless and sit rock-bottom of the Conference South.
"Anything is possible," claimed Adams. "We're one of the lowest ranked sides left in the competition but if everyone plays well and Weston have a bit of an off-day then I think we can cause an upset.
Bedford Town v Weston-Super-Mare
The FA Trophy
Third Round Qualifying
3pm, Saturday 29 November 2014
The Eyrie, Bedford Town FC
Winners receive £4000 from prize fund
"It's a massive game; they're two leagues higher but we're looking forward to it. It will be a serious test and the biggest game a lot of the lads have played in."
Despite Weston's poor start to the season, Adams believes they are much better than their league position shows.
"I watched them on Monday night at Boreham Wood and they were very impressive and passed it about well.
"I actually think they're in a bit of a false position; they're going to be alright in the league that they're in and it's going to be a massively tough day for us on Saturday."
Having started in the Preliminary Rounds of the competition, Bedford have played every fixture so far at home, which Adams think will impact on the outcome of this tie.
"I think home advantage will play a big part," he said. "I think the lads really enjoy playing at the Eyrie. When the home supporters get behind us it really makes a difference.
"Touch wood, our home form has been okay and it's definitely helped in the previous rounds. Whether it will help on Saturday against Weston Super Mare is another matter.
"Obviously they're going to be looking for some sort of reaction to getting rid of the manager which sometimes happens, players raise their game. It seems as if they've got another gear which they weren't previously using."
Bedford's Trophy run has coincided with an upturn in league form and Adams puts that down to the combination of experience and youth within his side.
"Our squad is very young but I felt the attitude lift before The Trophy run. There were six in the starting 11 under the age of 21 on Saturday and then the bench was all 20 and 21-year-olds.
"I signed a couple of experienced heads around six to eight weeks ago, who seem to bring a little bit more out of the younger lads and that's what I was looking for."
