Mechanical instabilities accompanying the demixing transition of semi-dilute aqueous poly(N-isopropylacrylamide) (PNIPAM) solutions are probed for the first time with Brillouin spectroscopy, densitometry and refractometry. The particular role of the elastic moduli and the mass density at this coil-to-globule transition followed by molecular aggregation is investigated. Even though the demixing transition of PNIPAM solutions is denoted as a volume phase transition, it turns out that this transition is governed by the elastic properties, instead of the volume properties. This is consistent with earlier findings made for the demixing transition in chemically cross-linked PNIPAM hydrogels. Above the demixing temperature, Brillouin spectroscopy discriminates compact PNIPAM-rich agglomerates with sizes larger than 200 nm. Interestingly, these agglomerates possess a sharp distribution of elastic moduli, which can be attributed without any doubt to a material with gel-like mechanical consistency. Thus the phase-separated PNIPAM-rich agglomerates are not in the glassy state.