Difference between F-test and separate t tests on each variable?

Why can't we separately test each variable(using t-test), whether or not it is equal to 0? Why should we use the F-test for checking if all coefficients combined are equal to 0?

To elaborate on the answer of @Erik, there may be an issue of multiplicity control with t-tests. Suppose (for simplicity) that none of the variables are related to the outcome, and that the variables are independent.

Then, when testing each variable with, e.g., a t-test at level $\alpha$, you will reject that null with probability $\alpha$. Hence, the number of $k$ rejections in $N$ tests is a Binomial random variable with probability mass function $$\Pr\nolimits_k=\left(% \begin{array}{c} N \\ k \\ \end{array}% \right)\alpha^k(1-\alpha)^{N-k}.$$

Take $\alpha=0.05$ and $N=20$. Then, the probability of at least one false rejection, the Familywise error rate, is given by $$FWER=\sum_{j=1}^{20}\left( \begin{array}{c} 20 \\ j \\ \end{array}% \right)0.05^j(1-0.05)^{20-j}=0.6415$$ Obviously, the problem only gets worse with $N$, so if you test many hypotheses at some fixed $\alpha$ each, you are almost bound to get at least one false positive.

Consider the case of one factor with many levels. For example, different cell line. The F-test can show you whether the dependant variable is influenced by cell line and the t-test can show whether two specific cell lines differ.

There are some issues with using only the t-values:

1. If you have coefficients, you will need to correct for false positives
2. Your regression coefficients depend on the reference level of your factor. If you set one of the extremes as the references you will get more significances.
3. It is possible that you have low power for each individual t-test, but high power for the F-Test. This is the case when you have many groups with small sample sizes.