We present a sample of 33 Balmer break galaxies (BBGs) selected as Hubble Space Telescope/F160W dropouts in the deepest CANDELS/GOODS fields (H ≳ 27.3 mag) but relatively bright in Spitzer/IRAC ([3.6], [4.5] < 24.5 mag), implying red colors (median and quartiles: < H-[3.6]> ={3.1}2.83.4 {mag}). Half of these BBGs are newly identified sources. Our BBGs are massive (< {log}(M/{M}ȯ )> ={10.8}10.411.0), high-redshift (< z> ={4.8}4.45.1), dusty (< A(V)> ={2.0}1.52.0 {mag}) galaxies. The spectral energy distributions (SEDs) of half of our sample indicate that they are star-forming galaxies with typical specific star formation rates (SFRs) of 0.5–1.0 Gyr‑1, qualifying them as main-sequence (MS) galaxies at 3 < z < 6. One-third of these SEDs indicate the presence of prominent emission lines (Hβ + [O III], Hα + [N II]) boosting the IRAC fluxes and red colors. Approximately 20% of the BBGs are very dusty (A(V) ∼ 2.5 mag) starbursts with strong mid-to-far-infrared detections and extreme SFRs (SFR > 103 M ⊙ yr‑1) that place them above the MS. The rest, 30%, are post-starbursts or quiescent galaxies located >2σ below the MS with mass-weighted ages older than 700 Myr. Only two of the 33 galaxies are X-ray-detected active galactic nuclei (AGNs) with optical/near-infrared SEDs dominated by stellar emission, but the presence of obscured AGNs in the rest of the sources cannot be discarded. Our sample accounts for 8% of the total number density of {log}(M/{M}ȯ )> 10 galaxies at z > 3, but it is a significant contributor (30%) to the general population of red {log}(M/{M}ȯ )> 11 galaxies at 4 < z < 6. Finally, our results point out that one of every 30 massive {log}(M/{M}ȯ )> 11 galaxies in the local universe was assembled in the first 1.5 Gyr after the big bang, a fraction that is not reproduced by state-of-the-art galaxy formation simulations.