I ran a regression with 4 variables, and all are very statistically significant, with T values $\approx 7,9,26$ and $31$ (I say $\approx$ because it seems irrelevant to include the decimals) which are very high and clearly significant. But then the $R^2$ is only .2284. Am I misinterpreting the t values here to mean something they're not? My first reaction upon seeing the t values was that the $R^2$ would be quite high, but maybe that is a high $R^2$?
The $t$-values and $R^2$ are used to judge very different things. The $t$-values are used to judge the accurary of your estimate of the $\beta_i$'s, but $R^2$ measures the amount of variation in your response variable explained by your covariates. Suppose you are estimating a regression model with $n$ observations,

$$
Y_i = \beta_0 + \beta_1X_{1i} + ...+ \beta_kX_{ki}+\epsilon_i
$$

where $\epsilon_i\overset{i.i.d}{\sim}N(0,\sigma^2)$, $i=1,...,n$. 

Large $t$-values (in absolute value) lead you to reject the null hypothesis that $\beta_i=0$. This means you can be confident that you have correctly estimated the sign of the coefficient. Also, if $|t|$>4 and you have $n>5$, then 0 is not in a 99% confidence interval for the coefficient. The $t$-value for a coefficient $\beta_i$ is the difference between the estimate $\hat{\beta_i}$ and 0 normalized by the standard error $se\{\hat{\beta_i}\}$.

$$
t=\frac{\hat{\beta_i}}{se\{\hat{\beta_i}\}}
$$

which is simply the estimate divided by a measure of its variability. If you have a large enough dataset, you will always have statistically significant (large) $t$-values. This does not mean necessarily mean your covariates explain much of the variation in the response variable.

As @Stat mentioned, $R^2$ measures the amount of variation in your response variable explained by your dependent variables. For more about $R^2$, go to wikipedia. In your case, it appears you have a large enough data set to accurately estimate the $\beta_i$'s, but your covariates do a poor job of explaining and\or predicting the response values.
