Was the Lord of the Rings trilogy the first film series to be shot concurrently?

The Three Musketeers

When Alexander Salkind and his son Ilya produced The Three Musketeers in 1973 they shot so much footage that they decided to split it into two movies: The Four Musketeers (1974).

This had ramifications and resulted in the Salkind Clause:

For their daring, the Salkinds have gone down in legal history:

• actors' agents and lawyers adopted the so-called 'Salkind clause', which prohibits producers from salvaging footage from the cutting room floor for a second movie without paying the cast accordingly.

This is also mentioned on the IMDb Trivia page:

As a result of the producers splitting the film into two parts, Screen Actors' Guild contracts now often feature what is called a "Salkind Clause," which requires producers to state up front how many films are being shot, and that the actors involved must be paid for each.

Superman

The Salkinds would later do something similiar with Superman (1978) and Superman II (1980), by shooting them back-to-back.

Although, for the sequel director Richard Donner was replaced by Richard Lester, who had previously directed the 'Musketeer' movies for the Salkinds.

From the Superman Wiki:

It was decided early in the process to shoot two films simultaneously.

During the production of 'The Three Musketeers' (1973), the Salkinds had realized that there was enough footage for two films and split the film in two, releasing 'The Four Musketeers' a year later.

The joint production of 'Superman' and 'Superman II' would mark the first time this process was used intentionally.

All actors' contracts have what is now known as the "Salkind clause", which stipulates how many films are being made.

All performers on 'Superman' were contractually obligated to 'Superman II' as well. However, in this case, most of the simultaneously-shot footage was reportedly scrapped when Richard Lester was brought in to finish 'Superman II'.

Kill Bill

Because of the Salkind clause new deals had to be negotiated with the actors when Kill Bill (2003) was split into two films in post-production.

In the case of a movie like "Kill Bill," which becomes two movies after the fact, SAG (Screen Actors Guild) requires a new round of bargaining for actors appearing in the second installment.

No, there have been several over the years, as detailed by a wikipedia entry on back to back filming.

Here's a short list, there are more in the wiki entry.

1. Back to the Future II and III
2. Kill Bill was filmed as one film and later split into two.
3. Superman the movie and Superman II (Earliest on the list, 1978 timeframe)
4. Deathly Hallows I and II were filmed together.