It depends on your modem and internet service.
A proper "cable modem" will only allow one device to be connected. If your router is having problems connecting to it:
- Try connecting the router, power cycle the modem, wait a minute, then power cycle the router. Sometimes the modem is programmed to "lock on" to the first device that's connected to it after it's been power cycled. So if you test the modem with your computer plugged into it, then swap your router for your computer, the router may not get a connection because the modem has already "locked on" to your computer.
- Other times, your ISP has to program which device is allowed to connect to the modem and you need to give them a call after you've plugged in the router.
There's another type of cable modem which I doubt you have, but does allow you to connect a switch. It's a type that allows you to have multiple static IP addresses. You're supposed to connect a switch, then assign your static IP addresses to your devices that will have Internet access. If you are not paying your ISP for a static IP, I highly doubt you have this type of modem.
Some cable modems are actually a combination modem/router. Usually these will also provide wifi service and have extra LAN ports, but not always. If you have one of these, yes you can plug a switch in and it will work.