Can I use the linear regression result with few independent variables? When I run the linear regression using about 12 independent variables, I get insignificant F-test result overall. 
So I discarded variables to make the F-test significant while having no multicolinearity problem checking by VIF test.
Then I come up with a linear regression with significant F-test and no multicolinearity problem.
However, I am left only with three independent variables and significant t-test results for these coefficients.
Shall I still use this result?
 A: For variable selection try using LASSO or ridge regression. Both of these perform variable selection. LASSO has the added benefit of zero out coefficients of insignificant variables.
Both are forms of penalized regression. The penalization parameter can be obtained with cross validation. 
All of this can be done with R, using the glmnet package and the glmer() and cv.glmer() functions.
Another approach is use a validation set to compare error rates from models or use area under ROC curves. It really depends on what you are trying to do.
As far as number of variables, as long as OLS assumptions are met, yes.
A: As whuber said, it's not really possible to recommend without more info. As already mentioned, make sure you really understand the assumptions ols regression makes, understand how to check such assumptions, and then you will be able to make the decision--and back it up. There are entire books on the topic as well as many online resources. Weissberg's Applied Linear Regression is fairly accessable start. 
