The Hawks had to settle for a 2-2 Blue Square Bet South draw at Weston-super-Mare after leading twice at the Woodspring Stadium.
Striker Ryan Moss twice blazed the Hawks in front in an exciting second half.
But the Seagulls levelled for the second time, with just two minutes remaining.
Even then the Hawks could have won the game with two late chances.
Full-back Dan Butler cracked a shot against the foot of the post and Sahr Kabba missed out in a one-on-one with the goalkeeper.
Manager Lee Bradbury felt it was a game his side could and maybe should have won.
‘It feels like a loss,’ said Bradbury.
‘It speaks volumes that we have come away disappointed not to have won the game.
‘We had chances to win it and it is hard to take because we didn’t get what we deserved out of the game.
‘It is another good point away from home but we could easily have grabbed all three.’
Bradbury’s side made a confident start, controlling the opening 20 minutes of the contest.
The bright start faded, though, and the Seagulls took charge for the rest of the first half.
On the half-hour mark, striker Chas Hemmings crashed a shot back off the Hawks crossbar.
And just before the break, Clark Masters turned another goalbound shot around the post at full-stretch.
‘In the first half we were off it a little bit,’ said Bradbury.
‘The players are finding it difficult to maintain the tempo I am asking them to play at in every game.
‘The longer we do it, the fitter they’ll get and that hopefully will bring greater consistency.
‘In the second half we did a lot better at pressing them higher up the pitch.’
It was pressure from Moss that led to the 51st-minute breakthrough after he harried home keeper Lloyd Irish into a poor clearance.
Steve Ramsey seized on the loose ball and fed Ollie Palmer, who slid the ball across goal for Moss to divert it in from close-range.
Weston levelled on 71 minutes, after Masters dropped a cross under pressure.
But Moss restored the visitor’s lead nine minutes later with a superb individual goal.
Receiving the ball on the edge of the Weston area, he turned past his defender and buried his shot high into the net.
The Hawks felt a little hard done by when the Seagulls struck a second time just two minutes from time.
Substitute Stephan Hamilton-Forbes seemed to have his heels clipped as he broke in midfield but the decision went the home side’s way.
When the ball was played forward, Hemmings picked his spot from just inside the area.
‘It is frustrating because we could have moved up a few more places with a win,’ said Bradbury.
Hawks: Masters 7; Hinshelwood 7, Butler 7, Bailey 6 (Hamilton- Forbes 45mins – 6), Harris 8, Woodford 7, Ramsey 7, Hutchinson 7 (Arthur 61mins – 7), Palmer 7, Moss 8 (Kabba 80mins), Taggart 7.