Board Thread:Questions and Answers/@comment-5767007-20140325061554/@comment-6053530-20140325071023

They only lose their chance to cast a counter within that counter chain; in other words, the turn player can start another counter chain by activating a "Counter" ability. Bear in mind, though, the previous counter chains must have all abilities resolved before starting this new chain.

Refer to page 9 of the detailed rules here for more information.