You don't say what form the bounding dates are in, so I'll assume Unix timestamp.
This might get what you're after:
$now = time();
if (($now >= $some_timestamp) && ($now <= $some_later_timestamp)) {
echo 'Now is between ' . date('F jS', $some_timestamp) .
' and ' . date('F jS', $some_later_timestamp);
} else {
echo 'Now is not between ' . date('F jS', $some_timestamp) .
' and ' . date('F jS', $some_later_timestamp);
}