you could try
$explode_brs = explode($your_string, "<br>");
$count_brs = count($explode_brs);
you would then have to subtract 1 to get the number of breaks
eg.
word<br>word<br>word
would give an array count of 3(0,1,2) by subtracting 1 you get the actual number of <br>
i havnt tried this so if it's wrong sorry!!