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Palpation utilizes the fact that solid breast tumours are stiffer than the surrounding tissue. However, cancer cells tend to soften, which may enhance their ability to squeeze through dense tissue. This apparent paradox proposes two contradicting hypotheses: either softness emerges from adaptation to the tumour’s microenvironment or soft cancer cells are already present inside a rigid primary tumour mass giving rise to cancer cell motility. We investigate primary tumour explants from patients with breast and cervix carcinomas on multiple length scales. We find that primary tumours are highly heterogeneous in their mechanical properties on all scales from the tissue level down to individual cells. This results in a broad rigidity distribution—from very stiff cells to cells softer than those found in healthy tissue—that is shifted towards a higher fraction of softer cells. Atomic-force-microscopy-based tissue rheology reveals that islands of rigid cells are surrounded by soft cells. The tracking of vital cells confirms the coexistence of jammed and unjammed areas in tumour explants. Despite the absence of a percolated backbone of stiff cells and a large fraction of unjammed, motile cells, cancer cell clusters show a heterogeneous solid behaviour with a finite elastic modulus providing mechanical stability.