Bradbury's play-off hopefuls were on course to steal a march on their rivals after Danny Ings's double had given them a 2-0 lead against 10-man Yeovil on Saturday.
But and not for the first time this season Cherries pressed the self-destruct button, conceding twice in the closing stages to fritter two potentially-priceless points.
Cherries were rocked when Shaun MacDonald's injury-time header added to Adam Virgo's penalty four minutes from time as the Glovers staged a dramatic recovery.
Earlier, Ings had netted from the spot and bagged a delightful second to put Cherries in the driving seat after Yeovil defender Paul Huntington had been sent off five minutes into the second half.
Bradbury, asked whether he had seen coming the fight-back, replied: No. But it happened and we have got to learn from it. It was immature and naïve of us. At 2-0 up, we should have seen out the game. They threw the kitchen sink at us at the end but we should have dealt with it.
When they went down to 10 men, it was almost as if we had gone down to 10. We took a backward step when we needed to grab the game by the scruff of the neck and show some killer instinct. At 2-0, we looked comfortable. But after they had made it 2-1, I was always worried they were going to get another.
Under both Bradbury and his predecessor Eddie Howe, Cherries have squandered numerous points during the final few minutes of games this season. And with three matches remaining, they are clinging to sixth place with a packing field just three points behind them.
Bradbury added: It is something we have got to address but I think it also comes with age as well. I was shouting for us to see out the game, run down the clock and put the ball in the corners. I can only shout on so much, the lads need to take a bit of responsibility themselves.
It was a very disappointing afternoon for us. After 80 minutes, you thought it was going to be a great day and then that happened. It felt like a loss. It is another game where we are still where we want to be but we should have had three points instead of one.
There were some really good individual displays. But, as a team, we need to mature and grow up very quickly because you don't get chances to win games like that too often. We need to learn quickly because I don't think we will get many more chances.
Cherries were forced into a late change after ever-present midfielder Anton Robinson withdrew from the squad after his fiancée Jessica went into labour before giving birth to a girl on Saturday.
Both Robinson and Jessica would like to thank the midwives at Bournemouth and Poole hospitals.
Robinson, who had last missed a league game on Boxing Day 2009, could return for today's clash with Bristol Rovers at Dean Court (3pm).