Since my post seemed to disagree with
danger i decided to look things up.
Wikipedia has an explanation and a chart which is rather complicated but I think it boils down to this. If I'm right, danger is correct that England can have a maximum of five and they treat CL winners different from Europa winners. England normally gets 4. Three of the four are always full entry, and one requires going in the qualification round. The top 3 PL teams have guaranteed full CL spots. If non-English teams win the CL and Europa then the 4th place PL team gets the qualifying round spot and that's that.
If an English team wins CL but doesn't finish top 4 in the PL, it takes the qualification round spot from the 4th place team, and England still gets 4. This is what happened with Chelsea and the Spurs.
If an English team wins Europa but doesn't finish top 4 in the PL, it gets an extra non-qualifying spot and England gets 5 total.
England also gets 5 if English teams win both CL and Europa and neither finishes top 3. England gets 5 spots, with the Europa winner skipping qualifying and the CL going into qualifying. Again the 4th place PL teams gets squeezed out.
It seems odd that Europa winner is treated better than CL winner, but I think that was done because nobody cared about Europa and they wanted to make it more worthwhile.
Also, the number of spots is recalculated every year. There was a chance Italy could take a spot away from England this year but England kept it when MCFC and Liverpool advanced and Juventus was eliminated