If you account for multiple hypothesis testing (which you should), this is true. To any regression analysis, you can imagine adding 1 million predictors which are really just random numbers. By chance, some of these predictors will have an association with the target, but to limit the number of false positives among so many hypotheses, multiple hypothesis correction (Bonferroni, FDR, etc.) is needed. Now, significant predictors among your original set are penalized for the additional 1M tests, and may no longer be significant. With a larger sample size, however, those real effects will persist, and with enough data, could survive the multiple hypothesis correction step and remain significant.
Adding more predictors requires a higher bar of evidence to declare any of them significant. For a particular effect size, this will require a larger sample size the more predictors you have.