If a CFA example shows a large t-statistic and a high p-value, should I reject the null hypothesis?

Not automatically. For the same test, same distribution, same degrees of freedom, and same tail convention, a larger absolute t-statistic should correspond to a smaller p-value. If the t-statistic looks large but the p-value is high, first check whether you are reading the same row, the same coefficient, and the correct tail.

Use this sequence:

If the p-value is high and correctly matched, you fail to reject the null. A common reason for the apparent mismatch is that the p-value belongs to a different coefficient or that a one-tailed test is being evaluated in the opposite direction.