Numerical simulation of two-dimensional bond percolation at the percolation threshold shows that configurations with no singly connected bonds (SCB) appear with a finite probability. Nevertheless, the average minimal width of the percolation channels is small. This result confirms an assumption used to derive the critical exponent of the critical current density in superconductor- insulator composites. Higher-order bonds are shown to play the role of SCB in configurations with small numbers of SCB. Implications of the properties of low-connectivity bonds on the physical properties of percolating systems are discussed.