I set my deflection limit to L/360, but the maximum deflection indicated in the summary of node displacements in PostProcessing shows a deflection of 1.5 inches. Isn't this above the limit that I set?

During steel design per the AISC ASD code, there are two types of deflection checks you can perform with STAAD. They are

1. Check for local deflection. This is usually applicable to members which are connected at both their ends to other members.
2. Check for the relative displacements between the nodes such as for a cantilever beam.

LOCAL DEFLECTION is defined as the maximum deflection between the 2 ends of the beam relative to a straight line connecting the 2 ends of that member in its deflected position.

If you go to

Help - Contents - Technical Reference - Commands and Input Instructions - Printing Section Displacements for Members

you will find a diagram indicating this is in figure 5.41.

To obtain more information on the difference between the 2 methods of deflection checking, please go to

Help - Contents - Technical Reference - American Steel Design - Design Parameters (which comes after Allowables per AISC code)

It will bring up section "2.4 Design Parameters"

At the end of the parameters table, you will see several notes. Please read Notes items 1 through 4 for the description of the two methods.

As you can see there, the default condition, which is also represented by a value of zero for the CAN parameter, is to perform the LOCAL DEFLECTION check.

Your question indicates that what you are looking for is a check of the nodal deflections. The cantilever style check STAAD offers is probably the solution for your problem. If so, specify the CAN parameter with a value of 1.