Turned out they could not. Brennan Johnson gave them a flying start with his fourth goal in eight games but Spurs fizzled and disappointed again at the home of West Ham.
Alexander Isak netted a brilliant solo effort in the first half before a late penalty by Dominic Calvert-Lewin earned Everton a 1-1 draw to reignite the fight against the drop.
KATHRYN BATTE AT THE LONDON STADIUM: Both teams had chances to take the lead, and while the game slowed, a late flurry threatened to break the stalemate.
BURNLEY 1-1 WOLVES - IAN WHITTELL AT TURF MOOR: Vincent Kompany's hopes of leading Burnley on an late-season survival charge was undone by Rayan Ait-Nouri.
Follow Mail Sport's live blog for the latest scores, team news and updates from Tuesday's Premier League action including Tottenham's London derby with West Ham.
CRAIG HOPE AT ST JAMES' PARK: They still haven't won since before Christmas but snatching a draw from a game they should have lost will feel like a victory on some level for Everton and Dyche.
The Manchester City striker has consistently hit the headlines for his goal-scoring prowess but the Norwegian has come under fire after failing to match the lofty heights of last season.
NOTTINGHAM FOREST 3-1 FULHAM - JOE BERNSTEIN AT THE CITY GROUND: Gibbs-White was the inspiration for Nottingham Forest's win.
BOURNEMOUTH 1-0 CRYSTAL PALACE - JAMES SHARPE AT THE VITALITY STADIUM: In the end, even a mystery illness failed to stop Antoine Semenyo from playing a key role.
