600+ amateur teams enter the FA Cup tournament and about every 7-8 years a professional, non-league side makes a run, sometimes as far as the 5th round when the prem comes in. For context, Football Association "League Football" consists of the Premiership (20), Championship (24), League One (24) and League Two (24), a total of 92 teams. However there is a way up and down for 5000+ teams nationwide below the "4th Division" starting at what is now called "Step 5": https://en.wikipedia.org/wiki/English_football_league_system
Millwall in 03/04 and Cardiff in 07/08 were Championship ("2nd division") sides to make the FA Cup Final in the Prem era. In the last season before the Prem (91/92), Sunderland were at the bottom of the old 2nd Division and made the FA Cup Final.
And while there have been no lower-division party crashers in the FA Cup Final for 15 years, Leicester winning the Prem in 15-16 more than makes up for it, paying 5000-1 for anyone that punted on matchday 1.