How to predict when the electrical infrastructure on my 1950s rental is going throw a breaker?
A breaker will trip according to its trip curve, at 40&deg;C (104&deg;F). In this condition, it should be able to carry 100% of the rated load. So if you have a 20 ampere breaker, at 40&deg;C it should be able to carry 20 amperes. It's often recommended (and sometimes required), to only load a circuit to 80% its capacity.  

In most cases, devices are listed by the amount of watts they use, rather than the amount of current they draw. Luckily it's fairly easy to convert back and forth, as long as you know the system voltage.  

Watts = Volts * Amperes, Amperes = Watts / Volts.

So a 20 ampere breaker on a 120 volt system, should be able to supply 2400 watts.  Applying the 80% safety margin, that's 1920 watts.  A 15 ampere breaker can supply 1800 watts at 100%, or 1440 watts at 80%.

To estimate how much power you're using, add the wattage rating of all the devices on the circuit.