When it shows Barney seeing Ted and Robin holding hands in the park, it is nighttime. But when Ted and Robin are holding hands in S8 E23 "Something Old", it is daytime.
Ted said that Pocahontas was with John Smith, but she actually married John Rolfe.
"The Bro Code" in the hotel room is actually the same book anyone can buy at a book store. A small label is seen saying "As seen on CBS's How I Met Your Mother". It's very small, but it's similar to the one at a book store.
While Ted and Barney are on the beach in the rain holding hands, you can clearly see Barney's microphone under his white shirt.