I've been in NYC biglaw for several years and can count on one hand the number of months I have had over 300. It's rough, so good job hanging in.
It can be really hard to turn down work, especially if you like to be a good team player, so here's what I suggest instead. Tell the partner/sr associate, "Sure, I can help out with that. I have deadlines X and Y on case Q and R at the moment though, so I cannot get to it until tomorrow/next week/whenever." If all your work is coming from the same case, just tweak the script a bit: "Sure, I can help out with that. I'm currently working on projects A and B for this case. I will wrap those up tomorrow afternoon and then I can start on your project C and finish it by the end of the week. Let me know if that works or if you want me to prioritize differently."
This puts the burden on them to decide just how urgent their project actually is. As a junior, you don't always have insight into what is truly urgent vs. what can wait. It also helps them figure out if they need to staff up the team, which can be helpful to lighten your load.
Finally, when you tell them when you can get to their project, be kind to yourself. Meaning, if you can get to their project tomorrow only if you pull an all-nighter tonight, then don't say that! Give yourself an extra day. Nobody will protect your time but you. This can get tough if the entire team is pulling all-nighters constantly or if you're up against a hard filing deadline - you don't want to be the associate who always logs off at 10pm while everyone else works until 2am.