An unconvincing 1-0 home victory over Cardiff on Sunday consolidated fifth place in the Barclays Premier League for Spurs, with head coach Sherwood acutely aware his side's own top-four aspirations are slipping.
Tottenham trail fourth-placed Manchester City by four points, but the newly-crowned Capital One Cup champions boast two games in hand over their rivals.
Spurs may have to overhaul north London foes Arsenal to squeeze into next season's Champions League, but Sherwood believes it is too early to rule out Everton, and maybe even eighth-placed Newcastle.
Tottenham travel to league leaders Chelsea on Saturday, with Sherwood hoping to load all the pressure on Jose Mourinho's men.
If Spurs cannot crash the Champions League party, Sherwood said they will at least try to spoil the European shindig for a few of their competitors.
"I don't know about Chelsea being more ambitious than us, we're going to the Bridge and that's definitely tough, but sometimes expectancy works against you," said the Tottenham head coach.
"Some of our fans might have been rubbing their hands expecting a few goals against Cardiff, but it doesn't work like that.
"You've got to earn the right for everything.
"Sometimes when the expectation is not with you, it makes it a little bit easier.
"Games against Chelsea at the Bridge, Arsenal at the Emirates and Liverpool at Anfield, perhaps the pressure's off us a little bit and we can cause a surprise.
"We're looking to mess up as many other people's seasons as we can, and enhance ours at the same time."
Roberto Soldado's first goal in nine games fired Spurs to victory on Sunday, helping to shrug off the dejection of a 1-0 league defeat at Norwich a week earlier.
Sherwood admitted the Cardiff clash was a tetchy, disjointed affair, but was happy to collect the three points and roll on towards another London derby.
"We're looking upwards, but we know we're not only chasing teams above us there's people chasing us up behind too," he said.
"All we can do is look after ourselves, we had a blip last week, we had a tough game in Europe and we've bounced back.
"It wasn't vintage and I'm not insulting anyone's intelligence here, it wasn't great against Cardiff: but then, you can't always be great.
"I would relish a challenge with Arsenal, but they would argue they were up chasing on Liverpool.
"I think Everton would still be in the frame, and Newcastle possibly will be in the frame.
"There are teams capable of putting a run together, so you just don't know.
"You can't ever guarantee you'll win this one, draw that one, and anyone can beat anyone.
"You don't win leagues and qualify for Europe by playing free-flowing, magnificent football every week.
"Sometimes you have to dig deep and dig out a result.
"I thought we did that on Sunday, but I thought we were relatively comfortable in the second period."