  In this work we revisit the question of whether Cosmology can be made
compatible with scenarios with light sterile neutrinos, as invoked to explain
the SBL anomalies, in the presence of self-interaction among sterile neutrinos
mediated by massive gauge bosons. We examine this proposal by deriving the
cosmological predictions of the model in a wide range of the model parameters
including the effective interaction strength $G_X$, sterile neutrino mass
$m_{\rm st}$ and active-sterile mixings. With those we perform a statistical
analysis of the cosmological data from BBN, CMB, and BAO data to infer the
posterior probabilities of the sterile self-interaction model parameters. BBN
mostly provides information about the effective interaction strength and we
find that $\log_{10}(G_X/G_F)\geq 3.6$ can describe the primordial abundances
at 95% CL. Our analysis of CMB and BAO data show that when allowing a wide
prior for the sterile neutrino mass its posterior is bounded to $m_{\rm st}\leq
0.91~{\rm eV}$ (95% CL) considering CMB data only and $m_{\rm st}\leq 0.37~{\rm
eV}$ (95% CL) when adding the BAO information. So the mass bounds are slighly
relaxed compared with that of a non-interacting sterile neutrino model but a
sterile neutrino mass of 1 eV is still excluded at more than $2\sigma$ CL.
Conversely if fixing the sterile neutrino mass and mixing to the values
prefered by short baseline data we find that while that CMB data alone favors
the self-interacting scenario, including the BAO information severly degrades
the agreement with the model. Altogether we conclude then that adding the
self-interaction can alleviate the tension between eV sterile neutrinos and CMB
data, but when including also the BAO results the self-interacting sterile
neutrino model cannot lead to a satisfactory description of the data.
