The slope and intercept t tests are distributed as central random variables when the 'true' slope and intercept are zero and the error terms are normal, independent and homoskedastic (there's probably a few assumptions I missed). They are distributed as non-central t random variables when the true values are non-zero (and the errors are normal, independent, etc). As pointed out in the comments, the regression coefficients are normally distributed under these assumptions.
Generally for large sample size, the (non-central) t is a very good approximation for the distribution of the regression test statistics. You can test the robustness of the approximation yourself by Monte Carlo simulation. Simply create random data (according to some model), perform linear regression, record the t-test statistics (as well as the slope and intercept), and repeat. Then look at the distribution of regression test statistics. If your generating model violates the assumptions of independence, homoskedasticity, normality, you will see how robust the approximation by t distribution is; you can also look at the robustness of the normal approximation for the regression coefficients.