(*Two cylinders that definitely intersect*)
c12 = {Cylinder[{{84.22237813658421`, 34.522546388753256`, 
      96.53944830048721`}, {35.78780363879622`, -1.795119903801833`, 
      89.4349045077725`}}, 3.4135672615053068`], 
   Cylinder[{{84.22237813658421`, 34.522546388753256`, 
      96.53944830048721`}, {-57.53876091362403`, -16.31678987297056`, \
-73.98748865769903`}}, 1.6763697603374625`]};

(*Yup they intersect*)
Show[Graphics3D /@ c12]

(*Now test whether they intersect with IntCyl and with int*)

IntCyl @@ c12 // AbsoluteTiming 
int @@ c12 // AbsoluteTiming