Slicks and a lock diff will do that...
It's all about the weakest link in the chain
In most cars with an open diff, this is the traction of a single wheel, which breaks before anything else (wheelspin), but means the drivetrain has less stress on it because there is less resistance to the rotational force at the other end (gripping the road). With an LSD or locked diff, you have effectively multiplied the resistance by two.
If you then strengthen the tyres with more grip, you shift more pressure onto other components in the drivetrain, such as the clutch. If the clutch slips, this means there is a limit to how much pressure is exerted on the axles or gearbox and the clutch is giving way first. Upgrade the clutch and this weakness disappears; suddenly the gearbox and axles have more pressure exerted on them...if the pressure is too great, one of these will snap.
Side note: Clutches are generally cheaper to replace than gearboxes, hence I often say there is such a thing as too heavy a clutch.