Based upon my own experience as reviewer (i.e., a sample of n = 1), yes.
Here is my experience: I review for several different different journals, but have reviewed extensively for one journal. This journal has acknowledged me as an Exceptional Reviewer and I have been asked to apply to be part of the Editorial Board. For this journal, the editors vote on membership to the editorial board. Additionally, editors are usually selected from the board.
The answer to your question depends upon your subfield and the journal for which you are reviewing (e.g., is this a commercial journal or a society journal; also what is the journal's organizationally structure and how are editors select?).
I would suggest focusing your reviews on this journal (i.e., decline reviews from other journals and accept reviews from this journal if you are limited for time) and possibly dropping the Editor-In-Chief (or head editor) an email expressing interest in joining the editorial board. If it is a societal journal (i.e., published by a scientific society), I would also make sure my membership was current and possibly start raising my profile in that society (e.g., serving on committees, networking at their meetings, etc).