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Assigning Common Weakness Enumeration (CWE) categories to Common Vulnerabilities and Exposures (CVE) records remains an important but largely manual step in vulnerability analysis. We study this task as a text classification problem and compare two modelling choices: a \emph{multi-class} formulation that predicts a single CWE per CVE and a \emph{multi-label} formulation that allows multiple assignments. Three transformer encoders (BERT Base, SecureBERT, and CySecBERT) are evaluated on three nested label spaces (83, 47, and 25 classes). Multi-class training achieves higher macro-F1 across all settings, although the gap to multi-label narrows from 21 to 2 percentage points as the label space shrinks. Post-hoc threshold optimisation on the multi-label side closes this gap on the 25-class setting. Confusion analysis shows that the dominant misclassification patterns follow the CWE hierarchy and are shared across all three encoders (Pearson $r > 0.92$), which suggests that the error structure is driven more by taxonomy design than by encoder choice. A hierarchy-relaxed evaluation that forgives within-family confusions raises macro-F1 from ${\sim}$81\% to ${\sim}$90\%, indicating that strict metrics understate branch-level classifier quality. CySecBERT achieves the strongest results overall, with statistically significant gains concentrated in the multi-label setting.