There (at least) three possible ways this can occur:
1) The difference between conditional and marginal effects in GLM. When you have a nonlinear link function (like logit), the conditional effects differ in interpretation and often in value from the marginal effects, even when the predictors are uncorrelated from each other. The marginal effect is the effect of a variable in the population (possibly with confounding removed). The conditional effect is the effect of a variable controlling for other variables. In linear regression, the marginal and conditional effects are identical when there is no confounding. In nonlinear regression, in general, they differ. This is also the problem of collapsibility. So even if there is no confounding, you might find that the conditional effect of a predictor is significant while the marginal effect is not and that their magnitudes differ even if there is no confounding.
2) Increased certainty due to eliminating residual variance. Add strong predictors into a model reduces the unexplained variation in the outcome, which decreases the standard errors of the estimates. It may be that by controlling for all the other predictors in the model, the predictor of interest finally becomes significant. Note that this would be true even if there was no confounding and even if the marginal and conditional effects were the same. There is more certainty about the effect of a predictor when it has been isolated.
3) Suppression. Suppression is when the conditional effect is larger in magnitude than an unconditional effect. This happens when two predictors both positively affect the outcome but are negatively correlated with each other, or when two predictors are positively correlated with each other but affect the outcome in opposite directions. For example, if admission into a program is based on having low qualifications (i.e., high need), but one's qualifactions and participation in the program both positively affect the outcome, then the estimated effect of program participation will be small when not conditioning on qualifications but larger when conditioning on qualifications. For another example, if anxiety and depression are correlated with each other but anxiety causes increases in activity and depression causes decreases in activity, the estimated effect of depression on activity levels will be large (and negative) controlling for anxiety but small or zero not controlling for anxiety.
It's hard to know which of the three (or others) is active in your data example. But it is certainly possible to observe the results you do. To make them more comparable, perform logistic regressions to measure all of your associations rather than t-tests from some and logistic regression for others. That way you can more easily observe the change in the magnitude and significance of the coefficients on the predictors.
