So, in orchestra we got given Bolero and the third trombone part has 236 rests before we come in (yes I counted). If you're wondering, I'm not some professional player, I'm the youngest in the group at 13 and the ages go up to around 18 (with a few exceptions) I've always counted by generally just listening to the changes in music, where from my experience, new set of rests are usually based on. I've tried using this strategy but it doesn't work well for this length. Anyone got any tips?
Listening for cue parts in specifically the Bolero is difficult, because of many repetitions and subtle changes, so I fully understand your difficulty, and I too get lost most of the times I listen to this piece. As others mentioned, I would expect the conductor to help your section with this problem, as he has the overview. But I can also offer some other tips:
1) Maybe understanding the structure of the piece would help you to be able to orientate yourself in the music. Maybe read this wikipedia article https://en.wikipedia.org/wiki/Boléro#Structure or other articles about the structure, and then listen to a recording of the piece and see if you can find cues you can use, especially in the instruments that are nearby to where you sit.
2) The very first theme on the flute starts after 2 bars, so if each round is 18 bars, then 236 minus 2 bars is 234 bars, which is exactly 13 rounds of the theme (of 18 bars).
3) Another great method I learned about counting many bars is to count the inside joints of your fingers with your thumb. Each finger (not including thumb) has 4 joints, so your four longest fingers times 4 joints makes 16 counts on one hand, which often coincides with the length of sections in some types of music.
Follow along with the music and maybe notate the score when a recognisable section comes up near to this part coming in, so you know when to count from.
A conductor should be making you aware of your part, but if they're not doing that then narrowing the time you need to count from to around 20-30 bars will certainly help. Maybe speak to your conductor (assuming you have one with an orchestra) and ask them to give you a cue to come in (or at least a cue to start paying attention)?
If you're a nerd: I was second oboe in a university orchestra, so I also had to count lots of rests. My brother taught me how to do it on my fingers: using binary counting, you can count from 0 to 1023 on your two hands. Takes a little practice but it's fun.
Conductors have a habit of stopping and restarting at some place in the middle of your 236 bars rest. One technique I use is to mark the absolute bar number where you rejoin the fray.
So if the conductor says "start from bar 223" then you count 223 2 3 4 224 2 3 4 etc until you hit the bar where you start playing again.
When I was in the band we used to make a game out of counting long rests together as a section, looking back and forth at each other to check that we were at the same count as everybody else.
But seriously, I would say learn to be patient and count. There are good reasons to take it upon yourself to come in on time. It's good for your concentration skills to try to keep track of where you are in the music. It will also up your game and improve your musicianship to consider the rests part of the music and include silence as part of your performance.
I don't know how your sheet music is set up, but you should have some multi-measure rests between rehearsal markings. Instead of counting 200+ measures of rests in a row (if that's what you mean by 236 rests), you can count the number between rehearsal markings. For example, say you have 18 measures of rest between two rehearsal markings, I would count 1,2,3 2,2,3 3,2,3 4,2,3 and so forth up to 18,2,3. Then start over with the next multi-measure rest group. Another thing you can do is take a pencil and write a note to yourself what is happening at each rehearsal mark. This will help your ears to listen for and identify the musical cues and help you in the case you get lost with your counting.
As others have suggested, the best way is to learn the structure of the piece and learn how many measures you need to count from a given cue.
Alternatively (or additionally), you can also learn to count very large numbers on your fingers, for instance by using one of the methods described in this video: