I can't imagine different counties changing at different times. What if you lived in one and worked in another??
I imagine that life could get quite confusing during the changeover period!
This quote from the site linked above illustrates it quite well:
Having rallied the general public's support, the Time Uniformity Committee's goal was accomplished, but only after discovering and disclosing that on the 35-mile stretch of highway (Route 2) between Moundsville, W.V., and Steubenville, Ohio, every bus driver and his passengers had to endure seven time changes!
If you think about it, there are still some time issues like this with interstate working, although to a smaller degree. Consider somebody who lives in New Mexico and works in Arizona, or vice versa. In winter his home time and work time are the same, but in summer there is an hour's difference.
I seem to remember Europe and the US always changing at different times, though. When I studied in France back in 1994, I can remember having to deal with a 5 hour difference for a while instead of 6 hours.
The ending dates for daylight saving had been similar for several years, but not the start dates (last Sunday in March in Europe, first Sunday in April in the U.S.). Maybe that's what you're remembering?