A special form of the Birkhoff interpolation problem is investigated. We prove an existence theorem for certain types of interpolation which, in a particular case, reduces to a theorem of Meir and Sharma for $(0,2)$ interpolation by $C^3 $ piecewise quintics. The method of proof enables us to obtain $L_\infty $-estimates for the error in interpolating smooth functions. These error bounds are shown to be sharp by means of a Baire category argument.