In space with 3 or more dimensions, there are infinitely many pairs of lines that are not parallel and do not intersect.
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No, never. Parallel lines are line that will NEVER intersect.
Parallel lines in the Euclidean plane do not intersect but all parallel lines in the projective plane intersect at the point at infinity.
parallel ============================================= Parallel. _____________________ _____________________ Those two lines will NEVER intersect.
Coplanar lines that don't intersect are parallel. Lines that are not parallel and do not intersect are skew lines.