I have been a special educator for over 20 years.  I have a license in K-12 special education and elementary education.

I attended a meeting where I was told that all special educators have 5 years to get degrees in 2 out of 4 academic areas that they teach.  

Is this accurate? Is there no provision to "grandfather" older teachers?

From Sue:

Teachers need degrees, or certification, or to meet some other state measure of competence in the subjects they teach. 

What you were told about the need to get a degree is too narrow, at least in the federal law. But your state or district may require more than the federal law.
 
According to federal law, (NCLB and IDEA 2004), elementary school teachers must be certified in elementary education. You are certified so you meet the highly qualified requirements for a teacher in an elementary school. You also meet the requirements for a special education teacher in grades above elementary school if all your students are being taught to an alternative standard that is not above elementary level.
 
Pam's article on the requirements for special education teachers covers all the aspects of it so I will refer you to that  http://www.wrightslaw.com/idea/art/tchr.hq.require.htm 
The section of the Elementary and Secondary Education Act of 1965 that the IDEA 04 refers to is the definition of a highly qualified teacher in No Child Left Behind. I have copied a summary of that below.