I've never seen an oil lease, although I have seen mineral leases. Ignoring the contract, you would have until the statute of limitations ran, which is somewhere in the range of two to four years, depending upon which state the property is located in. But more likely the damages are governed by the terms of the contract.
Complicating this is why aren't the lease payments being made. If the company is not paying as a result of fraud, the period would only start to run once the payee knew, or should reasonably have known, that they weren't being paid.
For example, the oil company sends statements to the land owner that nothing is being extracted. Land owner lives five hundred miles away from the property, and never visits it. Despite the statements from the oil company, it is extracting 5,000 gallons a day, starting 1 January 2000. On 1 January 2006, a neighbor lets the land owner know about the extraction. Land owner does nothing.
Land owner goes out to look at the land on 1 January 2010, and sees the pump in operation and then checks his next statement, which says nothing is being pumped. Land owner sues oil company. Statute of limitations in the state and contract limit damages to three years.
Because of the neighbor reporting the information about the pumping to land owner, the land owner would only be allowed to go back three years or until 1 January 2007.
However, if the neighbor did not report the pumping to the land owner, and it appears that the oil company was being fraudulent, then the land owner could go back to 1 January 2000.
But you'd do better if you tell us what the plot needs. We can construct a plausible scenario to make it work.
If this is a real scenario, contact an attorney. There are way too many variables to attempt to provide information based upon the limited facts you're giving us.
Best of luck,
Jim Clark-Dawe