Stimulated by a question raised in the papers of J. Scheuneman [Bull. Am. Math. Soc. 77, 589-592 (1971; Zbl 0223.53048)] and J. Milnor [Adv. Math. 25, 178-187 (1977; Zbl 0364.55001)], the author proves the existence of compact nilvarieties which carry no complete affine structure. For that, he constructs \(n(=11)\)-dimensional nilpotent Lie algebras with no faithful \((n+1)\)-dimensional linear representation.