Sometimes you'll see brownack discolouring on a connector or power transistor. At that point, you don't need schematics. The same goes for blown capacitors, no power, and incorrect status LEDs.
If you are troubleshooting the operation of the board, it's harder to do without schematics, but there's still stuff you can do. If you have a 2-layer board. You can hold it in front of a light and you will see the traces inside. If your board has one major chip, you can refer to the datasheet of that IC to understand more. I did this with an LCD interface board - I scoped some data lines to see if there was activity on it.
But there are times when you absolutely need the schematics. If the test points are not labeled with anything more descriptive than TP8, TP9 etc., then you to look up what they are.
I like playing devil's advocate here. There are a lot of repairs I must do without schematics. Knowing what the purpose of the board is helps me understand a lot, and let's me make guesses like “this big resistor must be the feedback one”. But often I will tell my supervisor I cannot do the repair without drawings. Most often, in these cases, I make measurements on the board, but they are useless because I don't know what nominal (ie correct) values should be.
My friend told me a ture story:
My first shift as a new technician at Texas Instruments. I had spent 6 years as a tech in the USAF, and during that time took an advanced electronics course aimed at senior technicians.
So coming in on my first day, I had the credentials, at least on paper.
The shift supervisor pointed to a switching power supply tester, with a non-functioning unit sitting on it, a schematic next to it.
First day, no training, no orientation on this strange machine, had never seen this power supply.
All he did was show me how to switch on the power to the tester.
There was available an oscilloscope, volt-ohm meter, other usual testbench equipment such as soldering irons and replacement transistors and other parts.
So looking at the schematic, I started with checking the emitter-base leads for the right voltages, other points on the board I could check for good or bad readings.
Within 15–20 minutes, I’d found the bad transistor, changed it out, showed the supervisor I had repaired the power supply board.
Because I had a schematic, and the knowledge to use it.