This article examines the free size-dependent vibration behaviour of an Euler–Bernoulli type double-microbeam system, made of two microbeams elastically connected, with simply supported, clamped–clamped and clamped–pinned boundary conditions using an assumed-mode expansion technique (AMET) along with finite-element verifications; effects of altering the couple-stress-tensor-related parameter and the stiffness coefficient of the elastic connection are studied. The double-microbeam problem is formulated nonlinearly utilising Hamilton’s principle, alongside the modified couple stress theory (MCST), and energy terms. Employing the AMET, the free vibration characteristics were obtained through solving motion equations. For the double-microbeam system and the elastic connection, a finite-element method (FEM) based model and simulations are developed which enabled us in a partial verification of our results; the agreement is excellent. Moreover, another set of partial verifications is performed with available data in the literature by neglecting one of the microbeams and the spring connection. The study revealed increasing either the couple-stress-tensor-related parameter or stiffness coefficient of the elastic connection increases the natural frequencies of the double-microbeam system. It is also found that both the microbeams vibrate in the same manner in the first mode expansion and out of phase by 180° in the second mode expansion. Altering the boundary effect from pinned–pinned to clamped–pinned, and again to clamped–clamped was found to increase the natural frequencies of the double-microbeam system accordingly.