It is well-known that peakons in the Camassa-Holm equation and other integrable generalizations of the KdV equation are H1-orbitally stable thanks to the presence of conserved quantities and properties of peakons as constrained energy minimizers. By using the method of characteristics, we prove that piecewise C1 perturbations to peakons grow in time in spite of their stability in the H1-norm. We also show that the linearized stability analysis near peakons contradicts the H1-orbital stability result for the Camassa-Holm equation, hence the passage from the linear to nonlinear theory is false.