The problem with designing on poor-quality displays is that every poor-quality display has different kinds of problems. I expect what you're talking about is designing on a panel with relatively small gamut, but there are other issues that are equally likely to affect the quality and fidelity of colours on a display:
- Some are fine when viewed directly in front but quickly deteriorate when viewed at an angle.
- Some have internal colour or backlight intensity variations which mean parts of the display look fine and other parts look terrible (or darker, or brighter, or undersaturated, or oversaturated).
- There are displays/configurations with very low colour temperatures and ones with very high colour temperatures, leading to very warm or very cool whites and other colours. Likewise some people mess with the gamma settings on their displays, setting them to very different levels to other users.
If you're targeting a specific device/hardware profile (as in mobile development), you really should check the output on the display itself (especially if the display uses a different kind of display altogether such as an OLED display, especially a PenTile one, since the colour reproduction can vary dramatically from a conventional LCD screen—especially saturation). There are some great tools to help you with that such as LiveView (for iOS devices) and Android Design Preview (for Android devices).
However if you're designing for general use across a very wide and uncontrolled set of devices there's no reason to suspect that designing on one terrible display will ensure better fidelity on some other bad display. It's my belief, then, that you should design on a well-calibrated, high-fideltiy display if possible and, as with any other design task, test the output in as realistic a condition as you can, on a few different displays of varying quality.
Pay special attention to the way JPEG artefacts are shown; designing on one display and choosing output settings/compression ratios in Photoshop to be unnoticeable can sometimes mean viewing on a machine with much lower gamma reveals all those terrible-looking compression artefacts plain as day.