Born in Camden, Bunn started his career with Tottenham Hotspur as a trainee but joined Northampton Town before he was offered a professional contract.

His impressive performances ousted long term first choice keeper Lee Harper, who subsequently left Sixfields for Milton Keynes Dons.

Following injuries to the other keepers at the club he was restored to the first team towards the end of the year and subsequently signed an extension to his loan deal, keeping him at Bramall Lane for the remainder of the season.

An injury to Robinson persuaded Blackburn Rovers to invoke an early recall to Ewood Park in March, with Bunn departing having played thirty five games for the Blades.

[16] He made his Premier League debut on 28 December 2010 in a 3–1 away win against West Bromwich Albion coming on as a half-time substitute for Paul Robinson who suffered an injury during the game at The Hawthorns.

On Wednesday 26 October 2011, he started in the League Cup fourth round against Newcastle United at Ewood Park, playing the full 120 minutes after Gael Givet scored the winner in extra-time with a header, the game finished 4–3 to Blackburn.

He made his full debut on 26 December 2011 against Liverpool at Anfield in a 1–1 draw, making a reflex save in the last minute to earn Rovers a point.

On 31 December 2011, just five days later, Bunn started and played the full 90 minutes in goal against Manchester United at Old Trafford in a stunning 3–2 away victory for Rovers.

On 29 August 2012, Bunn joined Premier League side Norwich City for an undisclosed fee on a two-year deal, with an option of a further year.

[17] He made his first Premier League appearance of the season on 24 November, replacing first choice goalkeeper John Ruddy due to injury in the 82nd minute in a 1–1 draw away to Everton.