Having failed to reach the final stages of any ranking event in the first three seasons of his career, Cavney won seven matches to qualify for the last 32 of the 1995 World Championship at the Crucible Theatre. He defeated John Bayliss 5–0, Roger Leighton 5–2, Andrew Cairns 10–6, Michael Judge 10–9, Les Dodd also 10–9, Nick Terry 10–8 and Neal Foulds 10–9, setting up an encounter with Malta's Tony Drago. In the event, Drago led 8–0 and although Cavney claimed the next two frames, the deficit was greater than the latter could recover, and Drago won 10–2.