C172 pilot reported radio problems while in the pattern at a non-towered airport resulted in a NMAC requiring evasive action.

Narrative

Flying from the south of ZZZ to enter the traffic pattern to land on Runway XX; I began making recommended radio calls at 10NM out; and then upon 45 entry to downwind; downwind; turn to base; turn to final. Except for at about the 10 NM out point as I recall; I never heard any other traffic in or around ZZZ though I could hear traffic at neighboring airports - mostly reporting positions in and around ZZZ1. On short final; another pilot reported that he had just been 'cut-off' so was executing a 'go around'. The only part of his aircraft identification I remember was 'Aircraft Y' and a reference in his transmission to the airport name of ZZZ1 but then he corrected himself to say ZZZ. I acknowledged that I must have been the aircraft that 'cut him off' and apologized. He stated that he had been making radio calls for the last 10 minutes and didn't I hear him. I stated that I had as well and asked if he didn't hear me as I was genuinely curious if he could not hear me. The other pilot never responded but only 'keyed his mic'. I am unable to provide an estimated miss distance as I never actually saw the other aircraft either before or after the incident. After taxiing to the parking area; I entered the FBO building and asked the employee behind the desk if he had been monitoring the radio and whether he heard the radio communication from either aircraft. The employee stated that he had been monitoring the radio but that it had been consistently unintelligible. While talking to the employee; an aircraft made a radio transmission that I could hear over the employee's handheld radio. It was not understandable due to static and the employee stated something like 'like that'. After a few minutes; I departed ZZZ to return to home base airport. As I was departing; I heard the Aircraft Y pilot again making radio transmissions and referencing ZZZ1. During the weeks leading up to this incident; other pilots of the aircraft I was flying reported concerns with the radio equipment. On the day of the incident; I conducted a radio check prior to departure from home base. On the 3rd attempt; another aircraft responded that they could hear me fine. I am uncertain if either of the two aircraft pilots operating in the pattern at the airport of departure heard my first two attempts at a radio check and just never responded or if the 3rd attempt was the first attempt that the responding pilot heard. On the day after the incident in question; the aircraft was flown by another pilot to a nearby towered airfield and reported that the Tower reported 'garbled' communication from the Com #1 and nothing on Com #2 though the pilot reported reception on Com #1 was 'fine'. The aircraft was left with an avionics technician at the towered airport to determine proper functionality of the radios.