He told his players and club chiefs of his decision immediately after the 5-1 defeat by AFC Wimbledon - the second home game in a row the Rocks had let in five.
Also leaving is Andy Awford, who arrived with Jenkins as number two in the summer, and two members of their backroom staff.
But coach Darin Killpartrick is staying and seems likely to be appointed manager, at least on a caretaker basis.
Jenkins said he was leaving with heavy heart but he felt he had taken the team as far as he could.
He told the Observer outside the dressing room: "I don't see the picture here improving and I have a certain amount of personal pride.
"We have put a lot of time and effort into trying to make things happen this season but it has not worked out as we had hoped.
"I don't like walking away from things and this is the first time I've done it in football management, and I'm not doing this lightly. But this will give the club a chance to look at their situation and maybe for someone else to come in and take over.
"It's been hard. Over Christmas it's been a miracle we've found enough fit players to get a team out.
"This was the second successive home game we'd lost 5-1 and it's not been a nice experience. Losing by that many to Havant on Boxing day was a dagger through the heart.
"Andy and I feel we have taken this squad as far as we possibly can."
Jenkins said the need to be more than team manager had taken its toll. Both he and Awford had also been using their contacts to bring in extra finance - with some success - and mixing this with running the team had been tough to juggle.
Jenkins and Awford arrived in June when the club announced they were able to operate on a similar playing budget to last season - around 3,000 a week for players and management.
But in August the clubhouse was ruled out of bounds after being damaged by fire, hitting income severely, and in November the committee was forced to make significant further cuts to the playing budget.
It left Jenkins with a squad over-heavy with young players and often unable to name five substitutes. There were just two on the bench against the Dons.
"The players have shown great spirit but what has shown more than ever in recent weeks is that the squad as a whole is not good enough to compete in this league," Jenkins added.
"The frustration for the management has been that we have not been able to go out and bring in the standard of players we have known we have needed.
"But I'd like to thank everyone at Bognor for the support they have given us. We are leaving on amicable terms, I hope, without bitterness, and I wish the club nothing but the best of luck for the rest of this season and beyond.
"We're leaving behind a good, honest set of lads in the dressing room. We got a certain way along the road with them but results weren't right and we feel this is the right thing to do."
Jenkins expects to take a break from football for a while, saying his hours spent trying to get things going at Nyewood Lane had started to affect his business life.
More on Jenkins and Awford's resignation, and on what the immediate future holds for the Rocks, on this website over the next few days and in the Observer on Thursday.
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