Whilst Linux distributions like DSL and Tiny Core are ideal candidates for running on older thin client hardware, newer thin clients come with more powerful processors and more RAM. Despite this they may well not be suitable for running the standard large Linux distributions such as Ubuntu or Fedora. For example Ubuntu uses Unity as the default GUI and the documentation states "In order to run Unity the system needs a more capable graphics adapter".
However these more powerful thin clients are perfectly capable of running the lightweight variants of the larger Linux distributions. There is no formal definition of lightweight in this context, but essentially they are distributions where the basic components (such as the desktop) have been chosen to minimise the demands on the underlying hardware. They're designed to run reasonably on low-spec PCs, or what were probably high-end PCs several years ago.
Examples of lightweight desktops are Openbox, LXDE, and XFCE.
If you are interested some words about system requirements and 'lightening' a standard distribution can be found on the Ubuntu documentation pages.
Installing and Tuning Linux provides a guide to selecting, installing, and tuning Linux Distributions for Thin Clients.
Using Crunchbang provides installation notes on installing Crunchbang to an IDE hard drive or Compact Flash card on thin clients such as the HP t5720 or Neoware CA-21.
Using LXDE discusses some of the LXDE distributions available and then concentrates on installing Lubuntu 12.04 on a Compact Flash card on a suitable thin client.
Using XFCE discusses installing the Ubuntu version of the XFCE desktop to an IDE hard drive or Compact Flash card on thin clients.
Ubuntu Issues outlines issues and work-arounds for installing Ubuntu 12.04 on thin clients with VIA C3 and Eden processors with Samuel and Nehemiah (non-PAE) cores.