We propose confidence intervals with near-nominal coverage of variance-covariance parameters in linear mixed models, even when interest or nuisance parameters are near the boundary, that is, when some random effects have small variances or are strongly correlated. In such settings, our simulations show standard Wald, likelihood ratio, and score intervals with nominal level 95% can have coverage as low as 45% and as high as 99%, while the proposed intervals are near nominal. The proposed intervals invert a modified profile score statistic based on the restricted likelihood, with nuisance parameters estimated on an extended parameter set. In a longitudinal study of socialization in children with autism, where the estimated correlation between the random intercept and slope is near minus one, our intervals show that children who start higher improve more slowly. In a genetic study of white blood cell count in mice, they distinguish the contributions of variant classes even though several intervals include zero. Compared to universal inference intervals, which are valid in finite samples, ours are 1.7 to 3.1 times narrower, and finite where one universal interval is not, at a forty-seventh of the computing time. We establish asymptotically correct coverage uniformly in the parameter under a condition known to hold with independent clusters and crossed random effects.

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