I hope you can find this useful.
The default clearances on a new PCB project are usually set to around 8 mills (0.2 mm), which is too tight to get a trace to connect to a pad on a tight pitch QFN package. Meaning the clearance of one pad will interfere with the trace as it connects to another pad.
Depending on which board house you chose to use for fabricating your PCBs, the clearance from one signal to another can usually be tighter that the default in KiCad. (Most board houses will specify a minimum space and trace)
You can change the minimum clearances in the Design Rules dialog box. If you use net classes, the clearance will need to be changed for each class. If you use the default class for all nets, you will only need to change the one. Clearances of 6 mills (0.1524 mm) worked well for me with the 24QFN package. The width of your trace will also play an effect on being able to connect, mine was at 9.84 mills (0.25 mm) at the time I got this to work.
Again, I hope this helps.