  We perform Bayesian model calibration of two-nucleon ($NN$) low-energy
constants (LECs) appearing in an $NN$ interaction based on pionless effective
field theory (EFT). The calibration is carried out for potentials constructed
using naive dimensional analysis in $NN$ relative momenta ($p$) up to
next-to-leading order [NLO, $O(p^2)$] and next-to-next-to-next-to-leading order
[N3LO, $O(p^4)$]. We consider two classes of pionless EFT potential: one that
acts in all partial waves and another that is dominated by $s$-wave physics.
The two classes produce broadly similar results for calibrations to $NN$ data
up to $E_{\rm lab}=5$ MeV. Our analysis accounts for the correlated
uncertainties that arise from the truncation of the pionless EFT. We
simultaneously estimate both the EFT LECs and the parameters that quantify the
truncation error. This permits the first quantitative estimates of the pionless
EFT breakdown scale, $\Lambda_b$: the 95% intervals are $\Lambda_b \in
[50.11,63.03]$ MeV at NLO and $\Lambda_b \in [72.27, 88.54]$ MeV at N3LO.
Invoking naive dimensional analysis for the $NN$ potential, therefore, does not
lead to consistent results across orders in pionless EFT. This exemplifies the
possible use of Bayesian tools to identify inconsistencies in a proposed EFT
power counting.
