I would be more comfortable answering this question if you provided more information. I think what you are saying is that the F statistic is not significant but given Y= a+B1+B2+B3+e you find a statistically significant t statistic on B1 and B3.
I'm then interpreting your question as whether you can make a valid statistical inference about B1 and B3 given a F statistic that is not significant.
The short answer is no.
In order to calculate a coefficient such as B1 you need to interpret the dependent variable Y at a statistically significant level which shows up in a significant F statistic.
See slide 7 here (http://monogan.myweb.uga.edu/teaching/ols/lm7.pdf) for the equation that calculates a coefficient. Notice the relationship with Y.
Does this help to answer your question?