We investigate the AlxCr0.2MnFe0.5Co0.3Ni0.5 (0.3 <= x <= 0.7) high-entropy alloys by combining experimental and theoretical techniques. X-ray diffraction and magnetization measurements indicate that Al alters the crystal structure and the entropy change upon magnetization-demagnetization while keeping the Curie temperature almost unchanged. First-principles calculations of the vibrational, magnetic, electronic, and configurational entropies show that the leading entropy change is due to the magnetic and vibrational degrees of freedom. The presence of the body-centered-cubic phase, showing a sizable elastic softening upon magnetic transition, brings about the substantial magnetocaloric effect in this family of alloys.