For those who think Obi-Wan may have misspoke, there is pretty clear evidence that in the current canon the Jedi have been around for a thousand generations, not just a thousand years. In The Clone Wars episode "A Test of Strength" a group of Jedi younglings are introduced to a droid named Huyang, who has been instructing younglings in lightsaber design for a very long time. Then two episodes later in an episode titled "A Necessary Bond", at 11 minutes 7 seconds in, Huyang says "If I may, sir, I have instructed younglings for over a thousand generations, and these are among the best I have ever seen."
Likewise, the canon novel Tarkin says on p. 115 that "Tarkin suspected that even if the Jedi had restrained themselves from rising against Palpatine in his moment of glory, the esoteric Order had doomed itself to extinction. Where their flame had burned bright for a thousand generations, technological might was the new standard."
The Clone Wars episode "The Mandalore Plot" also had some dialogue indicating there had been a previous "Old Republic" which had fallen, with Pre Vizsla brandishing an unusual black lightsaber design, referred to in Ultimate Star Wars as a darksaber, and saying to Obi-Wan "This lightsaber was stolen from your Jedi Temple by my ancestors during the fall of the Old Republic. Since then many Jedi have died upon its blade. Prepare yourself to join them!" So, Palpatine's comment in Attack of the Clones that "I will not let this Republic that has stood for a thousand years be split in two!" may simply have referred to how long the Republic had stood since being re-formed after a previous "fall".