Self-assembled copolymer micelles have been widely explored for numerous applications including cosmetic formulations and detergency, drug delivery, and agriculture. In many of these technologies at least trace amounts of surfactants and detergents are present, yet little is known regarding their effect on the copolymer micelle structure. In this paper we examine the influence of a nonionic micelle-forming surfactant, Triton X-100, on spherical, nonionic polymeric micelles composed of poly(butadiene)-co-poly(polyethylene oxide). Using cryo-TEM we find that relatively small surfactant concentrations (less than 1:1 molar ratio) are sufficient to disrupt the copolymer assemblies, and to yield, via dimerization, mixed polymer - surfactant micelles with characteristic diameters. Saturation of the polymeric micelles is reached with approximately 3 mM surfactant (1:8 mol ratio). Upon saturation, and in high surfactant excess, coexistence of two homogeneous micellar populations is found: saturated polymer - surfactant micelles, and much smaller micelles of pure surfactant. The lack of complete demicellization of the polymeric micelles is explained by packing constraints of the polymer hydrophobic chains by the added surfactant. This behavior is found to be characteristic of polymeric molecules with hydrophobic-to-hydrophilic molecular weight ratio close to, or exceeding, 0.75. We further found that structural transitions in polymer - surfactant mixtures are fast, and the systems reach equilibrium at time scales characteristic to the small molecule, in contrast with the slow equilibration in polymer - polymer mixtures.