Early in the film when Gogol's mother and father are sitting together and one of them is holding a letter that they plan to mail to India, there are 2 U.S. stamps on the envelope. These stamps are current (2007) stamps, yet the letters are being mailed in the late 1970's or early 1980's.
Several anachronisms show up in the 1977 flashback sequences
showing Calcutta (Kolkata); the railway station shows a sign for Indusind Bank which was not established until the late 1990s; there is a building of The Telegraph which was not launched till 1982; and there are shots of several bridges and buildings which are only recent additions to the city.
While in the International Arrivals area of JFK Airport, Gogol focuses on the holographic photos/art lining the hallway; this is a modern decoration of that area (and that technology wasn't prevalent ~20 years ago).
When the parents go to New York in 1977, the license plates on the cars are the current blue and white plates (with the state of New York in the middle). The color of the plates in New York in 1977 were orange with dark blue letters and numbers.
During a scene set just after the family has returned from spending the summer in India, the very distinctive bright yellow blooms of forsythias can be seen in their yard. Forsythias do not bloom in the autumn.
Right before the scene where the Gangulis are by the seashore and Ashoke and Gogol go towards the rocks, a 1987-1990 Chevrolet Caprice Station Wagon (really battered too) can be seen parked in the background, when the scene is set in 1977 (ten years before that model was even introduced to the U.S. market).
After the family reaches Kolkata by air they are shown traveling in a taxi on the Howrah bridge entering the city, but one does not have to cross the bridge to enter the city from the airport because both the airport and the city lie on the same side of the river Ganga.
The Gangulis' dark green Volvo 240 sedan has a New York license plate up front and a California one on the back, when the movie is set in New York and a registered vehicle in the state of New York must have a license plate from the state in which it is registered.
The vandalized mailbox: the flag being up normally indicates to the postman that there is mail to be picked up and is lowed by the postman when the mail is picked up. also, the pattern of the graffiti indicates it was put on first with the flag down; then the name was stuck on and the flag raised.