RTS is supplied by the DTE device and CTS is the response to the DTE. CTS can ususlly be configured to be up at all times, when CD is up, or to follow RTS. If the MODEM is configured such that CTS will follow RTS, then both of them being down would indicate a problem with the DTE. Resetting the MODEM probably would not affect this situation, so another answer is more likely.
If I had to guess, I would look to the MODEM as the source of the problem. I would proceed under the assumption that RTS was being delivered to the MODEM, but the MODEM was unable to function.
There are, of course many other possibilities, but that is my best guess. If the problem repeats itself, try this. Place a breakout box (BOB) between the DTE and the DCE. When the circuit fails, note the condition of the LEDs on the BOB. This will tell you what is going on between the two devices.
One other thing. CTS can be used for hardware flow control. A malfunctioning flow control system could be the problem as well.
Let us all know what you find.
