simulate this circuit – Schematic created using CircuitLab
Figure 1. LED flexible strip lights typically consist of three LEDs and series resistors wired between the positive and negative rails.
Each segment of these LED strips is wired as shown in Figure 1. You can cut anywhere along the strip, including through an LED, but it should be obvious from Figure 1 that cutting anywhere along the D1, D2, D3 chain will open-circuit that whole section. All the LEDs up to that section will light but the one, two or three left on that section will not.
Is there any way to have a length of LED lights that still work if I cut it at a random spot?
I think the above schematic and text should give you a good understanding.