Maximum number of predictors in regression Are there any rules about the maximum number of predictors in OLS regression given the number of observations?
I was argued that I should reduce the number of predictors (20) because I have 80 observations.
 A: There are no real rules to the ratio of variables to observations, as long as it is less than 1, theoretically, you should be fine. The math works out, and you get valid estimates.
What the person might be arguing about is that there are far too many variables that could not possible all be relevant in explaining the behavior of the response. Since you haven't provided the regression problem, let me give an example of my own.
Suppose you are doing a study to determine what affects the grades of high school students. You design a study where you collect over 100 predictor variables ranging from the dietary habits, ethical inclinations, family income, IQ level, gender, age, and many other things. However, you could only find 80 students to take your carefully constructed survey.
Now you have 80 observations and 100 possible indicators of why each students is different. You can't use all 100 because $p$ cannot be more than $n$ for the math to work out. Ca you use say 60? Sure! The math works, but will it make sense? 
Maybe it will, but most likely it will not because for all of 60 predictors to be relevant in explaining the grades of high school students, these 80 students will have to differ in all possible combinations of these 60 predictors. That is likely not going to happen.
Where do we draw the line? There is no hard and fast line. In general it is agreed to keep a parsimonious model where there are as few predictors as needed, but not too parsimonious that you fail to capture valuable variability. To strike this balance in a theoretically valid way, selection criterions like AIC/BIC or tests of Forward/Backward selection are used.
A: As you add more predictors you will find that the standard errors of your coefficients increase

@ARTICLE{altham84,
  author = {Altham, P M E},
  year = 1984,
  title = {Improving the precision of estimation by fitting a model},
  journal = {Journal of the Royal Statistical Society --- Series B},
  volume = 46,
  pages = {118--119},
  keywords = {glm, regression}
}

Before you undertake any form of selection of variables especially using an automatic procedure you might want to read

@article{babyak04,
   author = {Babyak, M A},
   title = {What you see may not be what you get: a brief, nontechnical
      introduction to overfitting in regression--type models},
   journal = {Psychosomatic Medicine},
   year = {2004},
   volume = {66},
   pages = {411--421},
   keywords = {glm, variable selection}
}

As the title promises it is a non-technical introduction.
