This is actually a true story.
While planing a study, my work colleague did a power analysis for an intended effect size and a priori power of 80%. It turned out that the required sample size was one hundred observations per one group (n = 100), so he needed two hundred overall (N = 200). By now everybody figured out that was about Student's t-test (you're right). Then, the colleague contacted some data collection company and asked them for 200 observations in his study. It was paid in advance.
What the company did was to accidentally collect two hundred observations per one group (n=200). Twice as much as it was indicated in power analysis. Probably they misunderstood or probably he didn't explain clearly. Anyway, they kindly apologized for the inconvenience and didn't charge him for the additonal data. But he was left with those four hundred (N=400), which according to power analysis created an overpowered study. He scratched his head and said that he was gonna randomly select one hundred observations in each group to retain intended power of 80%.
I'm not sure if what he planned was a proper choice, so I'd like to ask you what do you think of it all? Should he stick to his a priori power-analysis results and randomly select participants from already collected data? Lower alpha level below usual 5%? Or there are other options?
Besides it all seems to me as if more data is something bad.