A win would have been enough for South Africa to claim at least a top-two spot in their Super 12s group but they never really got going in their pursuit of 159 on a sluggish pitch at Adelaide.
The Netherlands, the lowest ranked team to reach this stage, pulling off a famous win allowed Pakistan to leapfrog South Africa to reach the semi-finals after beating Bangladesh later on Sunday.
Bavuma, who has registered just 70 runs in five innings with a strike-rate of 112.9 during his time in Australia, admitted he will likely consider his position as white-ball skipper in the next few weeks.
“It's been a tricky time," he said. “To be considering that now, a lot of it will be emotional in terms of my role as the captain of the team but it's something that I probably will think about.
“It would be emotional now if I thought about all of that. Whatever happens, I carry myself with dignity through all the good and bad times. If I were to leave, I would leave with my pride in tact."
The Netherlands amassed 158 for four after being asked to bat first, with Tom Cooper's quickfire 35 off 19 balls followed by Colin Ackermann's unbeaten 41 off 26 deliveries.
While all of South Africa's top-six reached double figures, Rilee Rossouw top-scored with just 25 off 19 balls and they reached 145 for eight to bring their campaign to an abrupt halt.
South Africa have in the past had to contend with accusations of 'choking' at previous World Cups and Bavuma admitted it is a label that will stick after this latest setback.
“I think it will always be there until we find ourselves in a situation where we get to a final and we come out on the right side of that result," he added.
“Everything was in our hands as a team. We had confidence and belief and the form behind us but we just couldn't do the business. We had opportunities and we just didn't take them."
The loss therefore signals a sudden end to Mark Boucher's tenure as South Africa head coach although he was more measured about whether his side badly lost their nerve.
“The more you don't do well in World Cups, I think it does start playing in your head a bit, I think that's just natural," he said.
“I don't think that's been the case of late, we've played in some tight games in World Cups and actually won them. In the past we've tended to lose those games.
“We were never really in this game here, to be fair. On paper, yes we should have won the game but the game's not played on paper, it's actually played out there in the middle."
Additional reporting by PA.