Radio and door chime intermittently working on my 2006 Cadillac CTS. Radio works then I exit the car and start it up again and radio not working or door chime?

Well its most likely a bad connection at the ignition switch a lose or corroded blade or female connector not a main power wire .Or a connection at the radio it self .
Problems like this ie .electrical problems seam hard to understand at first . But what you need to do is look at the problem 3 speakers intermittent yet they work sometimes the chime for the key same .
I would pull the radio fuse out of the fuse box first if the chime problem goes away the problem is likely a connection on the back of the radio or corresponding wiring connections .If not then you need to look at the ignition and door switch door open door closed .
In short don't ripe things apart its much cheaper and easier to test then form a plain . You may need to buy a cheap tester or meter have fun with it.

check your fuses under dashboard..

is the radio hot after 30 minites,if yes replace the radio.
blown fuse can indicate chortcut.

YOUR INTENNA SIGNAL IS NOT COMING IN GOOD,MAYBE LOOSE CONNECTOR.

Don't do that again! You will need to re-boot the on board computer system by entering a code. Call your nearest Cadillac dealer.

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You need to take it to the dealer. One of the system computers is failing. I do not know that model but it could be either the body computer or dash computer. Good Luck.

Did you replace it with a factory radio or an after market one?
If you did the after market one,it's not wired correctly.

check antenna cable connections at radio & antenna.suspect power antenna if equipped.

Something to do with the reception or antena system..
Or the tuner in the radio is failing..

Rate well ;o)

Its you antenna module above your headliner. It controlls your remote for keyless entry as well as your am/fm reception via an amplifier. GM part number is 12233252 for updated and revised version around $100 bucks online.