A pilot was unable to contact RSW TRACON on the assigned frequency and was later told that a military aircraft was launched to investigate a NORDO aircraft on an IFR flight plan with an assigned squawk.

Narrative

After departing on an instrument flight plan and IFR squawk code and several frequency changes; I was handed off to RSW Approach frequency 132.5. I checked in with RSW and heard a response (somewhat garbled) with altimeter setting. After a while passed; I was not hearing normal radio traffic. I tried several times to contact RSW on assigned frequency; but did not get a response. I then attempted to contact RSW on 126.8. After several attempts; RSW called; told me to contact them in five minutes (126.8 is RSW Approach published frequency). I then called MIA Center. I finally got a response from MIA. They were unsure who I was and told me to contact RSW 133.5; which I did. They responded; then shortly there after gave me a frequency change and I went on my way. Upon landing at my destination I was told by an airplane that was with me on separate flight plans that a Coast Guard jet had been scrambled to intercept an 'unknown NORDO' on instrument flight plan; squawking assigned transponder code (with mode C) transponder indicating interrogation replies with altimeter reply. RSW not responding and then telling me to call again in five minutes! Maybe I missed the frequency but RSW only exasperated the problem by not responding to repeated calls on their published frequency.