Classifying a solid starts with what occupies its lattice: ions, atoms linked by a covalent network, metal atoms, or molecules. The strongest connection within a molecular unit is not necessarily the force holding separate units together. Graphite is a useful boundary case: strong covalent bonds form sheets, while electrons delocalized within those sheets allow conduction; weak interactions between sheets allow sliding.
Brittleness also follows from structure. Shifting layers of an ionic lattice can bring like charges alongside one another, promoting fracture. Metallic bonding can tolerate layer shifts while delocalized electrons maintain cohesion. When classifying an unknown, combine observations rather than using a melting point alone: no solid conduction but conduction after melting strongly supports mobile ions in the liquid.