The answer is no. We cannot determine boundedness of a RV X based on knowing only the variance of X. Two simple examples:
Suppose we know that X has finite variance.
Let X ~ N(0,1). Then P(X > c) > 0 for any real number c even though \sigma^2 = 1. That is, no matter how large a c we choose, there is a nonzero probability (p > 0) that X will be greater than c. So there exists no M such that P(X < M) = 1 a.e. So X is unbounded.
Let X ~ U(0,1). X is bounded.
In the reverse direction:
A RV that is bounded will have finite variance. This can be easily proved. A random variable that is unbounded may have finite or infinite variance.