No. 

Consider three variables, $X$, $Y$ and $Z = X+Y$. Their covariance matrix, $M$, is not positive definite, since there's a vector $z$ ($= (1, 1, -1)'$) for which $z'Mz$ is not positive.

Population covariance matrices are positive semi-definite.

(See property 2 [here](http://en.wikipedia.org/wiki/Covariance_matrix#Properties).)

*Sample* covariance matrices - [depending on how they deal with missing values](http://en.wikipedia.org/wiki/Missing_data#Partial_deletion) in some variables - may or may not be positive semi-definite. If pairwise deletion is used, for example, then there's no guarantee of positive semi-definiteness.