I used ESRI to create a point feature class using linear referencing, loading the data into an SDE database on SQL server. The resulting data set would not display in QGIS. I checked the SRID, validated the geometry, and narrowed the results.

I have seen issues with Line feature classes in QGIS where the line has measure values in ESRI. These wont display. Could this be the same issue with these points?

Yep. Of all the stupid things for ESRI to do, the points have measures.

Here are the properties of the point feature class:

Feature Type: Simple
Geometry Type: Point
Coordinates have Z values: No
Coordinates have measures: Yes

By including the measures in the points, ESRI is forced to provide a value for the Z when adding the records to SQL. This is due to a requirement that data is loaded in X,Y,Z,M where each value is space delineated.

So an example point looks like this:
POINT (1199100.2299588919 880237.32388755679 NULL 4.2200000000011642)

There is no way to add the measure without adding a placeholder for the Z.

As for the lines, I am going to have to say that it is a bug to not display correctly in QGIS for the same reason.

UPDATE: Thanks QGIS contributors! This is now fixed to handle this data.

Finally, to examine your data you can use these MSSQL methods:

Shape.AsTextZM()
or
Shape.ToString()