PA-28 Cherokee/Archer/Dakota/Pillan/Warrior
Flight Crew Instrument; Flight Crew Flight Instructor; Flight Crew Flight Engineer; Flight Crew Commercial; Flight Crew Multiengine
Flight Crew Last 90 Days 60; Flight Crew Total 2200; Flight Crew Type 200
Situational Awareness; Communication Breakdown
ATC Issue All Types; Conflict NMAC
General None Reported / Taken
Airport; Human Factors
We were practicing the RNAV (GPS) approach.  We were also told to report the IAF then a final report of a 2 mile final for the active runway.  The issue we had was while we were on final at 1300 MSL (700 AGL); an aircraft crossed about 200 ft below us from right to left. They were on right base; but overshot the extended centerline of final.  We noted the traffic to ATC with no additional action done of anyone's part.  No evasive action would have been beneficial.  We continued our descent on the RNAV approach.  I lost track of if they turned a wide final or where they went after we crossed over the top of them.  However; there were 2 aircraft that landed immediately after us.There were numerous aircraft in the pattern and holding short ready to take off.  This was the first day that the airport reopened after being closed for construction on the runways.  This created more radio communication length with the word 'shortened' following every transmission with the runway number.  Under normal circumstances; the volume of traffic is nothing unusual for ATC to handle.  Due to radio traffic taking longer; we ended up being about 1 mile past our requested reporting fixes before we were able to key the mike for a transmission.  The high volume of radio traffic; congested pattern; hazy horizon (typical Midwestern summer day) and runway construction are reasons that I believe we never made visual contact with the approaching Cirrus on right base.  I'm not sure if they saw us or not.  Perhaps it was a failure on both our parts for lack of visual scanning.  To mitigate this type of incident in the future; I will continue to scan for traffic and even request to depart the pattern if I feel safety is compromised or sense that ATC is being overloaded and having difficulty with traffic in the pattern.
PA28 instructor pilot reported a NMAC with an aircraft in the VFR pattern while executing an RNAV GPS approach. The airport traffic pattern was heavy and communication was congested.