.:random rocketshippiness:.
History
A Rocketshipper is someone that believes that Jesse and James are in love. It is combonation of the words "Rocket" and "relationship". (Get it? It's a pun- rocketship!) The term was invented by a friend of a fanfic writer named Rocket Jesse.
The term Rocketshipper has spawned many other terms. "Shippy" is commonly used to describe those moments (you know what I'm talking about) between Jesse and James. "Neoshipper" was invented for those that would like to see the newer Rockets, Butch and Cassidy, get together. Many other 'shipper terms have been invented, such as Pokeshipper, Gymshipper, Palletshipper, Egoshipper, Leagueshipper, etc. There are now literally hundereds of different ships.
Non-anime evidence
I could go on and on about evidence in the manga (comics), but all that needs to be said is that in Toshiro Ono's manga series, Jesse and James get married at the end, and Jesse is pregnant.
Yup, this is the pic from the Surf's Up Pikachu graphic novel. Cute!
People may say, over and over and over, "the manga is not the anime", but one important thing to remember is that Ono had to get approval directly from the writers of the anime for every single thing he drew and wrote in the manga. I doubt the creators of the anime would have approved this pic if they were against Rocketshipping!
Also, in the video game Pokemon Yellow, they always appear together. Just goes to show you how inseperable they are. They also appear together on the Here Comes Team Rcoket CCG card.
It's very rare that you ever see one without the other. They belong together!