When does a projective curve contain a planar subcurve?

If a curve contains a planar subcurve of degree r then its general hyperplane section contains a subscheme of degree r spanning a line. Here we study to which extent the converse is true. If r is small we describe counterexamples. However, we give an affirmative answer if r is large with respect to the degree of the curve. This result is achieved by studying curves which admit an irreducible two-dimensional family of r-secants. We show that such curves are forced to contain a planar subcurve of degree r unless they contain certain multiple lines. We describe the exceptional curves and give examples of them.