Brand New Bicycle Brake Cables Popping out
Some of the problems I encountered when doing this recently:


forgot to make sure the brake handle wasn't a little bit squeezed as I did it
forgot to use the brake handles fine tuning to increase the tension after installation
when cutting housing, forgot to use a pin to unblock inner sheathing at the end of the housing (the cable then pierces the end and catches)
I too had ill-fitting caps; I made some by wrapping narrow strip of duct tape until I got the right thickness
I forgot to oil the inside of the housing
I failed to cut the housing long enough to always have a decent bend creating pressure to keep itself jammed in