The mechanical properties of polymer composites, reinforced with silica-coated multiwall carbon nanotubes (MWNTs), have been studied using the nanoindentation technique. The hardness and the Young's modulus have been found to increase strongly with the increasing content of these nanotubes in the polymer matrix. Similar experiments conducted on thin films containing MWNTs, but without a silica shell, revealed that the presence of these nanotubes does not affect the nanomechanical properties of the composites. While carbon nanotubes (CNTs) have a very high tensile strength due to the nanotube stiffness, composites fabricated with CNTs may exhibit inferior toughness. The silica shell on the surface of a nanotube enhances its stiffness and rigidity. Our composites, at 4 wt % of the silica-coated MWNTs, display a maximum hardness of 120 +/- 20 MPa, and a Young's modulus of 9 +/- 1 GPa. These are respectively 2 and 3 times higher than those for the polymeric matrix. Here, we describe a method for the silica coating of MWNTs. This is a simple and efficient technique, adaptable to large-scale production, and might lead to new advanced polymer based materials, with very high axial and bending strength.