Louis van Gaal's side began the tournament with a stunning 5-1 thrashing of reigning world champions Spain but fell at the semi-final stage, going down 4-2 on penalties to Argentina after playing out a goalless draw in Sao Paulo.
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"I know I had a good game but, in the end, you stand here with your hands empty," he told NOS.
"It hurts so much. Our dream has burst. I've done so much to get here."
Wesley Sneijder, who also saw his spot kick turned away by Romero, echoed Vlaar's grief.
"I don't really know what to say right now," the veteran midfielder lamented. "This hurts, especially when you're one of the people to miss."
Netherlands must now pick themselves up for the third-place play-off against hosts Brazil in Brasilia on Saturday.