  We present a sample of 33 Balmer Break Galaxies (BBGs) selected as HST/F160W
dropouts in the deepest CANDELS/GOODS fields ($H\gtrsim27.3$~mag) but
relatively bright in {\it Spitzer}/IRAC ($[3.6],[4.5]<24.5$~mag), implying red
colors (median and quartiles: $\langle H-[3.6]\rangle=3.1^{3.4}_{2.8}$\,mag).
Half of these BBGs are newly identified sources. Our BBGs are massive ($\langle
\log(\rm{M}/\rm{M}_\odot)\rangle=10.8$) high redshift ($\langle z\rangle=4.8$)
dusty ($\langle \rm{A(V)}\rangle=2.0$~mag) galaxies. The SEDs of half of our
sample indicate that they are star-forming galaxies with typical specific SFRs
0.5-1.0~Gyr$^{-1}$, qualifying them as main sequence (MS) galaxies at $3<z<6$.
One third of those SEDs indicates the presence of prominent emission lines
(H$\beta$+$[OIII]$, H$\alpha$$+$[NII]) boosting the IRAC fluxes and red colors.
Approximately 20\% of the BBGs are very dusty ($\rm{A(V)}\sim2.5$~mag)
starbursts with strong mid-to-far infrared detections and extreme SFRs
($\rm{SFR}>10^{3}\,\rm{M}_\odot/yr$) that place them above the MS. The rest,
30\%, are post-starbursts or quiescent galaxies located $>2\sigma$ below the MS
with mass-weighted ages older than 700~Myr. Only 2 of the 33 galaxies are X-ray
detected AGN with optical/near-infrared SEDs dominated by stellar emission, but
the presence of obscured AGN in the rest of sources cannot be discarded. Our
sample accounts for 8\% of the total number density of
$\log(\rm{M}/\rm{M}_\odot)>10$ galaxies at $z>3$, but it is a significant
contributor (30\%) to the general population of red
$\log(\rm{M}/\rm{M}_\odot)>11$ galaxies at $4<z<6$. Finally, our results point
out that 1 of every 30 massive $\log(\rm{M}/\rm{M}_\odot)>11$ 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.
