A simple binary matroid, viewed as a restriction of a finite binary projective geometry $PG(n-1,2)$, is $I_{1,t}$-free if for any rank-$t$ flat of $PG(n-1,2)$, its intersection with the matroid is not a one-element set. In this paper, we show that the simple $I_{1,t}$-free and triangle-free binary matroids have bounded critical number for any $t \geq 1$.