I'm completely stomped. The picture below is how I usually set my breadboard and all the connections of a bjt.

When I try to measure the collector-emitter voltage, the voltmeter gives me a reading of the Collector-Emitter power supply (5V) instead. What am I doing wrong here?

I'll be measuring op amps on an integrated chip, so I'll definitely need to get this problem figured out.

 

In your drawing, the C-E voltage and the source voltage are one in the same. Try adding a load resistance between the collector and the source voltage.

 

Like this:

 

My instructor told me it was fine to have it setup without resistors as long as Base voltage doesn't exceed 1.1V.

Does it matter how big the power supply is to the Collector-Emitter, as long as it is sufficient to drive the BJT when Base-Voltage is at max (saturation)? Which leads into my next question.

I did give the resistors a try just like in your setup. I kept Base-Current constant, changed the Collector-Emitter power supply voltage, and then measured VCE. Turns out after a few different supply voltage, my VCE measurements capped out at approximately .7V. This did not seem normal as I was trying to get get a characteristic curve by plotting Collector-Current against VCE.

Appreciate the help.

 

how do you know that the base current is constant?

You should gave some resistor in the base circuit to provide some predictable current through the base.
In the setup you have, the base current is limited only by internal resistance of the 1V source and it can by quite high.

And when the maximal CE current is base current times beta, you may not reach the limit.