Back then most local stations only transmitted in the daytime. At night when radio wave propagation was unstable, having so many stations would jam the airways so you couldn't hear anything. However there were stations licensed as "Class 1-A Clear Channel" stations that had exclusive use of their frequency 24 hours a day. A few:
KMOX ST Louis
WSM Nashville
WLS Chicago
KNX LOs Angeles
KYW Philadelphia
The link below has a list of all of the clear channel stations:
http://en.wikipedia.org/wiki/Clear-channel_station