  Recent experimental efforts were made to explore the Bose-Einstein
condensation in multi-$\alpha$ system using the heavy-ion collisions. This
Letter provides a explanation why no signatures were observed and a guidance
for future experiments. More precisely, a harmonic oscillator model is
developed to study the multi-$\alpha$ source at or near zero temperature. The
temperature and particle population of the multi-$\alpha$ sources observed in
the current experiments are extracted. It is found that almost no $\alpha$
particles occupy the ground state in those multi-$\alpha$ sources. The critical
temperature for the multi-$\alpha$-condensed states are predicted, which
provides a guidance for the future experiments to probe the Bose-Einstein
condensation of $\alpha$ clusters in the heavy-ion collisions.
