  We study metric measure spaces that admit "thick" families of rectifiable
curves or curve fragments, in the form of Alberti representations or curve
families of positive modulus. We show that such spaces cannot be bi-Lipschitz
embedded into any Euclidean space unless they admit some "infinitesimal
splitting": their tangent spaces are bi-Lipschitz equivalent to product spaces
of the form $Z\times \mathbb{R}^k$ for some $k\geq 1$. We also provide
applications to conformal dimension and give new proofs of some previously
known non-embedding results.
