I have kept bone broth in my fridge for over 2 weeks as long as a layer of solid fat is at the top and intact. Never had it go 'off' yet, but as someone said there are various influencing factors.
I have a beef broth and a chicken broth in the fridge right now. The beef is going on 2 weeks and has three marrow bones and trimmings from several roasts and steaks in it. The chicken has only been going two days now. I bring them to a boil every 2-3 days, add some more veggies and meat trimmings, eat a little (usually with some miso), then pop them back into the 34º 'fridge. I keep them going until the pot is so full of bones there isn't any more room for veggies. Works like a champ.
Tonight my wife yelled at me because I skimmed off most of the chicken fat to make a broth sauce for breakfast; she wanted to use it for HER recipe tonight. Snooze you loose!
Lots of factors - was it stored in the coldest part of your fridge? Did you salt it? Reduce it? Those things can extend the shelf life. 3-4 days is typical food safety protocol, but I'll cop to using it up to a week if I had salted it and it passed the "smell test." Keep in mind most food safety recommendations are going to be pretty conservative and are very objective to minimize the odds of illness (idiot-proof, if you will).
Use within 2-3 days. By the 3-4th day, whats unused needs to be boiled once more, where whatever isn't used immediately might be good for another 2 days. After that, throw it out.
Which is basically a pain to do cool downs the first much AND then again the second time.
My advice? Invest in a pressure canner, make a whole bunch all at once, then pressure can it to where it can keep for a very long time.
As for telling when it goes bad - your nose knows. If its been 4 days already and you aren't sure, well, you might decide you are braver than I.
I keep it in the fridge for up to 4 days if I have made a small amount and know I'll use it up in that time. Otherwise I ladle it into jars (leaving a good gap at the top for expansion) and freeze it.
I couldn't bear to have any go off and have to throw it, after all those hours in a slow cooker!
For almost two decades I've routinely kept broth and other foods in the fridge past 2 weeks and have yet to become ill. Trust your senses... visual and smell test. If it smells good, eat it, if you're concerned, boil it or re-heat (nuke/bake) it.
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