There are several rules of thumb out there about how many subjects that you need for a multiple linear regression model. Most of these rules look at the ratio of subjects per variable (SPV). If you have 100 subjects and 20 independent variables in your regression model, then the SPV is 5. This article comes to the surprising conclusion that an SPV of 2 is just fine. In other words, you could have 40 subjects and 20 independent variables and still be okay. This is independent of power considerations, by the way, but it still seems rather small to me. Read the paper yourself and let me know what you think.
Austin PC , Steyerberg EW. The number of subjects per variable required in linear regression analyses. Journal of Clinical Epidemiology 2015, 68(6):627-636. Available at http://europepmc.org/abstract/med/25704724.