Fission fragment mass ratio distributions have been measured for the reactions ¹⁶O + ¹⁹⁴Pt and ²⁴Mg + ¹⁸⁶W, both leading to the same compound nucleus, Rn-210, at near-barrier energies. The measured fission fragment mass ratio variances for ¹⁶O + ¹⁹⁴Pt and ²⁴Mg + ¹⁸⁶W are compared with the calculations assuming compound nucleus formation. Mass variances of fragments from the ²⁴Mg + ¹⁸⁶W reaction show a dramatic deviation from the compound nucleus behavior. This suggests strong evidence for onset of the quasifission process in this reaction.