Bognor had to settle for a point as they made the long trip to Margate less than 48 hours after visiting Enfield.
They started brightly and in the first minute Dan Thompson chased down a pass from James Crane, forcing a save from Bull in the Margate goal.
With the home defence still unsettled, Ollie Pearce chased a through ball from Ashley Harris, only to see his attempt deflected for a corner.
This seemed to fire the hosts into a more positive mood, and indecision in the Bognor defence allowed Charlie Allen to get in a shot on goal bringing a fine save from Grant Smith.
A long cross from Wassamer on the right found Bodkin out wide and his cross from the left was met by Allen, who shot wide.
Only another good save by Smith and a bad miss by Jefferson Louis denied the hosts the lead.
On 16 minutes, probably against the run of play, Bognor took the lead when Pearce tapped in a great cross from Harris after a precision pass by skipper Kane Wills had freed the Pompey loanee.
Within two minutes Margate were almost level when, with the wind behind him, Matt Bodkin from the half way line almost chipped Smith in the Bognor goal.
The Bognor defence were really up against a powerful Margate forward line and it was no surprise on 25 minutes when the normally reliable Stuart Axten made a howler of a clearance giving the ball away to George Artemi, who rounded Smith to score the equaliser.
This seemed to give Bognor a bit of a kick up the backside and they got themselves back in the game.
Wills was looking fitter than at Enfield on Thursday night, and was taking more control in the middle of the park, while Doug Tuck was providing nice touches.
Another great pass by Wills saw Harris shoot wide from a good chance, and minutes later Thompson thought he was through on goal but was offside.
Margate had the final say in the half, but Louis shot wide.
With the strong wind now behind Bognor, they started the second half in determined fashion. Almost immediately another pass by skipper Wills set up Scott Chamberlain but he failsd to hit the target, and minutes later Pearce seemed to be through again only to be held back by Nelson - and the resulting free-kick led to nothing.
Artemi went through one on one but Smith denied him again.
Bognor replied with a good shot by Chamberlain forcing a corner which was headed wide by Craig Robson.
Terry Dodd repalced Harris on 62 minutes and he was almost involved in a goal following a goalmouth scramble, which the hosts scrambled clear.
Sammy Igoe replaced Thompson on 73 minutes but neither side was able to make that killer pass.
The draw saw Bognor drop out of the top five and they need to return to their winning form when they resume their league campaign at Grays in Thursday and at home to Wingate on Saturday.
BOGNOR: Smith,Hinshelwood, Crane, Axten, Robson, Tuck, Harris, Wills, Thompson, Pearce, Chamberlain. Subs: Igoe, Wollers, Sackman, Weir, Dodd.
See the first report from the Sussex Senior Cup semi-final against Horsham YMCA on this website this evening (Mon April 7).