The two-sided Simes test is known to control the type I error rate with bivariate normal test statistics. For one-sided hypotheses, control of the type I error rate requires that the correlation between the bivariate normal test statistics is non-negative. In this article, we introduce a trimmed version of the one-sided weighted Simes test for two hypotheses which rejects if (i) the one-sided weighted Simes test rejects and (ii) both p-values are below one minus the respective weighted Bonferroni adjusted level. We show that the trimmed version controls the type I error rate at nominal significance level alpha if (i) the common distribution of test statistics is point symmetric and (ii) the two-sided weighted Simes test at level 2alpha controls the level. These assumptions apply, for instance, to bivariate normal test statistics with arbitrary correlation. In a simulation study, we compare the power of the trimmed weighted Simes test with the power of the weighted Bonferroni test and the untrimmed weighted Simes test. An additional result of this article ensures type I error rate control of the usual weighted Simes test under a weak version of the positive regression dependence condition for the case of two hypotheses. This condition is shown to apply to the two-sided p-values of one- or two-sample t-tests for bivariate normal endpoints with arbitrary correlation and to the corresponding one-sided p-values if the correlation is non-negative. The Simes test for such types of bivariate t-tests has not been considered before. According to our main result, the trimmed version of the weighted Simes test then also applies to the one-sided bivariate t-test with arbitrary correlation.