Early on, when Francesca is riding with Robert in his pickup and is helping him get to the covered bridge he can't find, they pass a tractor pulling an implement. Played in slow-motion, the implement is a big round baler which wasn't invented until the early 1970's. Also, the John Deere tractor that is pulling it has decals on the hood with Doppler-type striping. This design wasn't used on their tractors until around 1980.
When Francesa and Robert are standing in the kitchen one night, the kitchen window over the sink shows it is dark outside. The camera swings to a shot of the refrigerator and in the background is a window but instead of being dark it is daylight outside.
Near the end of the movie, when Francesca and her husband drive up to the general store, the driver-side window is rolled up. When she leans over to see Robert's truck across the street, the driver-side window is completely rolled down.
When the kids find the Celtic cross that Francesca had given Robert, the engraved letters were clearly visible and barely worn, if at all. However, during the flashback of when Francesca first received the necklace after Robert's death, and well before the kids could have ever seen it, the name "Francesca" is very worn away by the years and is barely visible.
Near the start of the film, when Robert and Francesca first drive to the bridge, a shot of a rather modern-looking house appears behind. When the camera cuts back to Francesca a few minutes later, the same house/shot is used
When Francesca is driving with him in his truck to initially show him where the bridge is, there is no sunshine on Robert's shoulders. Suddenly, in the next shot it's on his right shoulder, then the next shot switches to his left and then back again.
After Francesca's husband and children return from the Illinois State Fair, her husband indicates it took them about 3 1/2 hours to drive back. That would barely get them from Madison County to the Illinois border. The Illinois State Fair is held exactly 374.28 miles from Madison County, at least a 5+hour drive in 1965.
Near the beginning of the movie a banker is depicted at the farm house having brought a safe deposit box with him to the house. This would never happen, safe deposit boxes never leave the general area of the bank vault and the banker would not have access to the box anyway. (Also, the term is NOT "safety deposit." The proper term is SAFE Deposit - the box-holder DEPOSITS their belongings in a SAFE).
When Francesca is listening to the radio, the aria "Mon coeur s'ouvre à ta voix" from the opera "Samson et Dalila", can be heard. The aria is sung by Maria Callas. This recording however was never approved for release during Callas's lifetime. In the summer of 1965 Maria Callas was still alive.
When they went to the bar (the place off the interstate) the first time they show the band on stage playing the bass player is a white guy and he is playing a old style upright bass. Moments later the bass player is now a black guy playing a Fender electric bass guitar.
However some jazz bands do use two bass players at the same time, specially one upright bass and one electric bass, so this may very well be incorrectly regarded as goof.
After Richard dies, Francesca tries unsuccessfully to find Robert via the National Geographic office. The box she received from Robert's lawyer upon his death had Robert Kincaid stationery clearly showing his address: PO Box 634, Bellingham Washington, 98225. Even his phone number: KL-5-8763. The letter she received from him in 1965 (the one in the safe deposit box) probably used the same stationery. Francesca could have hired a private investigator if all these avenues failed, especially as she knew he was from Washington state from his license plate.
Francesca took a big chance giving Robert's "4 Days Remembering" photography album to Lucy Redfield. If she had pre-deceased Francesca, it might have been lost. Francesca's children might never have received it. Much better to have shown it to Lucy during their friendship and then place it in the safe deposit box or the hope chest with all the other memorabilia.
Heavy rain is obviously artificially generated when Richard and Francesca stop at the grocery store toward the end of the movie. The visibility in the background (trees, houses, etc) is too clear for such a heavy downpour.
Also, when they are driving in the rain, at times it appears that sunlight is visible (which technically is possible, but unlikely given how overcast it appears in previous shots).