Unraveling this mess may be difficult. Seeing the 30 amp breaker join the two sides indicates that the two hot lines go straight down each side in the back of the panel rather than alternating every other breaker. If it were alternating, then any 220 circuit only needs two vertical slots rather than taking up all the space of both sides as well. What this means is that these two circuits can be joined anywhere since the lines are in phase with each other.
The easy fix is if you have an incorrect connection in that breaker panel, rather than each line going its separate way. If that's the case, you remove the connection and the problem is solved. So as others say, the first step is the open it up and take a photo (with the power off and being careful to avoid the lines feeding the main breaker).
The difficult fix is if the connection is somewhere else in the house. Unless you know how the home is wired, you are left with breaking the connection at various points until you find the location that roughly splits the circuit in half. To do this, you open up an outlet in one of the rooms, disconnect the load side of the outlet (pick a side if you don't know which is which), and see if only half the outlets turn on when you reset one breaker. Try to make the split in the circuit between two rooms.
Once you decide where to split the circuit, you're left with an extra wire from the other circuit in the J box to cap or remove. The proper solution would be to remove it, but doing so will likely require opening up the wall. If you cap it, make sure each wire cannot touch anything else to avoid the risk of a short, and then label the wire so no one hooks it back up in the future. I would also go through the extra effort to find the other side of this connection and disconnect it and label it on both sides.
Edit: Seeing the separate red and black wires makes me wonder if they continue through each outlet with two circuits like mgb suggests. If so you just need to locate where the two lines cross. If the two circuits are separated in the outlets themselves, then it should be obvious when you open an outlet up and see a red and black going to the separated top and bottom. In that case, turn the power off to both circuits, disconnect an outlet, cover the exposed connections, and turn the power on to only one circuit. If you now have two separate circuits, then then crossed connection is in the outlet you disconnected or one of the dead outlets in the home. If the circuits are still joined, then the problem is in one of the live outlets. Use the process of elimination to avoid opening too many outlets.
As always, when working with electricity, ensure the power is off before you risk anything touching a wire. Test ever wire in the outlet before beginning work, especially in your situation. And when you open the panel, the line side of the main breaker is still hot even when that breaker is shutoff. If you feel nervous about doing any of this yourself, then hire an electrician.
One possible investigatory tool is a "toner" set - Fox & Hound or some other brand - While they are are more often used on telecommunications cables, some do claim to work on power wires. User reports vary, possibly due to different toner sets working more successfully than others with power wires. They work by one part applying a small electrical signal to the wires, which the other part of the set can pick up without having to actually touch the wires. If you can get a tone on a circuit (Disconnect it at the breaker before hooking up the toner - assuming you understand the risks of and precautions to take when working in the breaker panel [scratch that - it's the landlord's problem - leave it alone]) and then you find a place in the wall where tone stops, rather than going off in any of the possible directions at that point in the wall, it might mean you've found the break. Check both sides of the wall, etc. to increase the odds of success. Try connecting the toner at one of the dead outlets and tracing backwards, as well. If they both stop in the same place coming from different directions, that's a good place to knock a hole in the wall.
Before any more fail, map whatever circuits still work, and perhaps turn off any you can't find out what's connected to (paying attention to things that might not be instantly obvious like furnace or boiler, well pump if any, sump pumps, etc...)
Best Answer
Given the locations for these breakers and source of trips, I'm guessing they are actually AFCI breakers. AFCI are designed to detect an arc fault, e.g. a failing extension cord that is at risk of causing a fire. They are required in bedrooms in newer electrical codes. Vacuums frequently trip these because the motor itself gives off a lot of noise in the line that resembles arcing (or perhaps there is some arcing inside the motor).
Since you are seeing it now, look for anything that may have changed, like your holiday lights. Try leaving those lights off for a while to see if the trips stop, and if so, inspect the wiring. Twinkling lights would be another suspect as they turn on and off with a heated element that is likely generating a small arc as it does so.