We calculate the trispectrum ${T}_{\ensuremath{\zeta}}$ of the primordial curvature perturbation $\ensuremath{\zeta}$, generated during a slow-roll inflationary epoch by considering a two-field quadratic model of inflation with canonical kinetic terms. We consider loop contributions as well as tree-level terms, and show that it is possible to attain very high, including observable, values for the level of non-Gaussianity ${\ensuremath{\tau}}_{\mathrm{NL}}$ if ${T}_{\ensuremath{\zeta}}$ is dominated by the one-loop contribution. Special attention is paid to the claim in J. Cosmol. Astropart. Phys. 02 (2009) 017 that, in the model studied in this paper and for the specific inflationary trajectory we choose, the quantum fluctuations of the fields overwhelm the classical evolution. We argue that such a claim actually does not apply to our model, although more research is needed in order to understand the role of quantum diffusion. We also consider the probability that an observer in an ensemble of realizations of the density field sees a non-Gaussian distribution. In that respect, we show that the probability associated to the chosen inflationary trajectory is non-negligible. Finally, the levels of non-Gaussianity ${f}_{\mathrm{NL}}$ and ${\ensuremath{\tau}}_{\mathrm{NL}}$ in the bispectrum ${B}_{\ensuremath{\zeta}}$ and trispectrum ${T}_{\ensuremath{\zeta}}$ of $\ensuremath{\zeta}$, respectively, are also studied for the case in which $\ensuremath{\zeta}$ is not generated during inflation.