They played nearly half of yesterday's defeat with nine men, and, just as they thought they had gouged a point out of a sapping, wild ordeal, they lost to a stoppage-time Bournemouth goal.
Tottenham's task was made all the more hard two minutes before the interval when forward Son Heung-min was sent off for pushing Jefferson Lerma and halftime substitute Juan Foyth lasted 125 seconds before seeing red for a lunging tackle.
Mauricio Pochettino's men may now require a result against Everton four days after a trip to Amsterdam as their involvement in the Champions League both this season and next remains uncertain.
Either side of those saves, Son should have scored after he waltzed through the defence but he fired over.
Tottenham's end-of-season dip has continued as the Lilywhites slipped to a disappointing 1-0 defeat to Bournemouth, and Son Heung-min's red card summed up a frustrating week for Spurs.
After Jefferson Lerma appeared to step on Son's foot after he fouled Fraser, the South Korean forward reacted and pushed Lerma over and received a straight red card from Craig Pawson to reduce Spurs to 10 men for the second half.
What does it mean?
Travers became the first teenage Premier League goalkeeper to start a match since Joe Hart in October 2006 and he soon showed why he was deserving of such faith. Pawson should have dismissed the erratic Dier and then awarded a penalty against the same man, while Son's push to the chest of Lerma was foolish and needless.
According to Opta, Foyth was only on the pitch for 2 minutes and 13 seconds before he was sent off.