How to filter a field in the add content window in Drupal?
You can try the Conditional Fields module. It allows you to show/hide fields based on the value of other fields.

The Hierarchical Select module suggested in jimajamma's answer is meant only for hierarchically structured vocabularies. Conditional fields doesn't have a fancy animated interface like HS, but it provides more flexibility. For instance, you could make a text field depend on a taxonomy term: