The Bantams are still without a League One win in six matches but twice came from behind at Valley Parade to deny the Sky Blues a fifth straight success in all competitions.
The game sprang into life with four goals in the opening half an hour with the visitors racing into a 2-0 lead, only to be pegged back straight away.
A mix-up between Bradford goalkeeper Jon McLaughlin and Rory McArdle allowed Andy Webster to give his side a second-minute lead, and another error from McArdle five minutes later presented Leon Clarke with the chance to extend Coventry's advantage - which he took.
However, the Sky Blues' defence was was also prone to errors as several missed clearances allowed Wells to pounce, putting his side back in it after 17 minutes, and then he went one better, levelling in the 28th minute after racing on to James Hanson's flick-on.
But Bradford undid all their hard work as the half-time whistle approached, allowing Callum Wilson time and space to squeeze a powerful shot in between McLaughlin and his near post three minutes before the break.
Things calmed down a little in the second half but there was late drama after Jordan Clarke handled the ball in the box. Wells stepped up in the fifth-minute of added time to complete his hat-trick and earn the home side a point.
The result sees the Bantams climb a place to seventh while Coventry remain 12th in the table.