This is something really interesting. I must say I have never seen this before. How come t-test of every single coefficient is significant while F-test of all the coefficients are non-significant?
For small sample size, it's possible that t-tests of some coefficients are significant while F-test is not. In that case, dropping those non-significant variables will make F-test significant.
If predictors are correlated, standard errors of coefficients will be larger, so it's possible that t-tests are non-significant while F-test is significant.
Maybe you post your data so we can play with it.