If $\hat\alpha=0.4$ with a corresponding $t=1.2$, you cannot reject $H_0\colon \alpha=0$ at the usual significance levels of 5% or 1%. Thus you do not reject the CAPM. (Your interpretation is fine.)

If $\hat\beta=?$ with a corresponding $t=?$ that is close enough to zero, you cannot reject $H_0\colon \beta=0$ at the usual significance levels of 5% or 1%. However, the CAPM says that $\beta\neq 0$, so you would actually want to reject $H_0\colon \beta=0$ if you were rooting for the CAPM. (Your interpretation is not fine.)

Failing to reject $H_0$ indicates that either you do not have enough power (the sample is too small) or the systematic risk (the explanatory variable) is actually irrelevant for explaining the expected value of return on an individual security (the dependent variable). Or perhaps some of the model's or estimation method's assumptions are violated, e.g. the systematic risk has been measured with error (which is quite probable), so we cannot fully trust the result.

Note also that we could try to formulate the null hypothesis as $H_0\colon \beta\neq 0$ but it would be very inconvenient to test it, so we typically avoid such null hypotheses.