The dynamics of single droplets in a bounded shear flow is experimentally and numerically investigated for blends that contain one viscoelastic component. Results are presented for systems with a viscosity ratio of 1.5 and a Deborah number for the viscoelastic phase of 1. The numerical algorithm is a volume-of-fluid method for tracking the placement of the two liquids. First, we demonstrate the validation of the code with an existing boundary integral method and with experimental data for confined systems containing Newtonian components. This is followed by numerical simulations and experimental data for the combined effect of geometrical confinement and component viscoelasticity on the droplet dynamics after startup of shear flow at a moderate capillary number. The viscoelastic liquids are Boger fluids, which are modeled with the Oldroyd-B constitutive model and the Giesekus model. Confinement substantially increases the viscoelastic stresses and the elongation rates in and around the droplet. We show that the latter can be dramatic for the use of the Oldroyd-B model in confined systems with viscoelastic components. A sensitivity analysis for the choice of the model parameters in the Giesekus constitutive equation is presented.