From the FA website:
"Since the war, no non-League club has got beyond the Fifth Round of The FA Cup.
The five clubs who have managed to reach that round are Colchester United (1948), Yeovil Town (1949), Blyth Spartans (1978), Telford United (1985) and Kidderminster Harriers (1994).
Spartans, now in the Conference North, hit the headlines as a Northern League side in the 1977-78 FA Cup. They accounted for Burscough (Cheshire County League) 1-0 at home, Chesterfield (Division Three) 1-0 at home, Enfield (Isthmian League) 1-0 at home, and Stoke City (Division Two) 3-2 away on the way to a Fifth Round meeting with Third Division Wrexham at the Racecourse Ground. A last-gasp equaliser by the Welshmen set up a replay at Newcastle United's St James's Park home nine days later.
The Sixth Round draw had thrown up "Wrexham or Blyth Spartans v Arsenal", but the non-Leaguers missed out on a dream tie, an amazing crowd of 42,000 watching them lose the replay 2-1.
Harriers took their place in The Cup's last 16 in the 1993-94 season after beating Kettering Town (Conference) 3-0 at home, Woking (Conference) 1-0 at home, Birmingham City (Division One) 2-1 away, and Preston North End (Third Division) 1-0 at home. The Conference side then faced West Ham United at Aggborough Stadium and were undone by a Lee Chapman goal."
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However, weren't Tottenham Hotspur a Non League club when they won the FA Cup back in 1901?