Spurs would say no.
It depends on what the clubs around you are doing. You may be able to have a good season and get into a european place but if a rival gets taken over by Opec and pushes you out it can be difficult to fight back in, regardless of how you are being run. Everton also saw the flip side, as St Chalet stated above, 4th just gets you into the qualifying stages, if you have no European pedigree you will face a harsh tie due to the ceedings. Everton couldn't get past a Villareal team in full flight. If it was Saints having a storming upturn in fortunes we would be in the same boat, facing a Valencia, A Steaua Bucharest, a Benfica.
The FA Cup win is a glory day, something to remember forever. A fourth place finish only gives you the certainty of a Europa League place.