I had fallen on someone elses property and broke my wrist
As the other attorneys have answered, the statute of limitations for an injury from a fall on someone else's property is two years, UNLESS, you fell on public property, in which case the statute of limitations is one year. There are other important issues that you need to consider. The law in Illinois protects landowners in many situations. The landowner is not necessarily liable for just because you fell on his property. You really need to talk to a local personal injury attorney and go through the particulars of how you fell, where you fell and what made you fall.
Law Office of Steven A. Sigmond
345 N. Canal #1208
Chicago, IL 60606
The 'time limit' you refer to is called a statute of limitations. Its highly advisable that you check with an experienced Illinois-based personal injury attorney, (I am not licensed in Ohio) but it appears (based on some quick Googling) that the statute of limitations in Illinois is two years from the date of the injury.
If you fell on property owned by a Municipal Corporation, the Statute of Limitations in Illinois is one year. Additional quirks apply. If you fell on private property, it's two years unless you are a minor at the time of your fall in which case you have until your 20th birthday at the close of business within which to file suit.
One way or another, get a lawyer. It's too easy to trip up.