DP left his last job because of this.
I can't remember how many were in the team, but management had a "max 2 people off at a time" rule. All the rest of them booked their leave and their package holidays at the start of the year in January. I could never do this, because my leave year doesn't start until April, and by then, all the non-school holiday times in the spring and summer were booked by his colleagues.
By the time we'd ruled out all the weeks where I had team meetings, mandatory training and stuff, we'd be left with a week in October and a week in February, not the ideal time to be touring the UK in a motorhome!
The stupid thing was that his job was very self-contained and quite specialist. No-one used to cover his work if he was off, he'd just come back to a backlog, and he didn't cover anyone else's or take phone calls or anything. It would have made no difference whatsoever if he was the 3rd person off in any given week.
Now I'm not doing frontline work, I don't have to do mandatory training, I'm in a team where we get very little urgent stuff and half the team can be off at a time and he's changed jobs to one where his leave year is the same as mine. For the first time in years we're going away for a week at the end of March to use up our leave, and I'm really looking forward to it.
OP, the one thing I would do is send your leave requests by email, so you have a record. You're entitled to 20 days paid holiday a year (plus 8 bank holidays), and if you're consistently being prevented from taking it, you can point this out and have the evidence to back it up.