Solution: Choose C. Reason: Using a regular pentagon as a counterexample can negate conclusion A; using a square as a counterexample can negate conclusion B; using an isosceles trapezoid as a counterexample can negate conclusion D. Therefore, the correct conclusion can only be C. In fact, to affirm C positively, it still needs to be proven that $nA_{1} A_{3}+A_{1} A_{n}$ leads to a contradiction. Therefore, the number of sides of a convex polygon with all diagonals equal cannot be $\geqslant 6$, i.e., it can only be a quadrilateral or a pentagon. Examples of a square and a regular pentagon show that convex quadrilaterals and pentagons with all diagonals equal do exist.