While this is not my area of expertise, it just happens to be that of my beloved girlfriend.

While there is a lot of speculation of when these two movies take place, we can be certain of one thing: Frozen and Tangled are connected due to the Rapunzel and Flynn cameo in Frozen. If we go by the most accepted theory that Anna and Elsa's mother and father was on the way to Rapunzel and Flynn's wedding when they died then we can assume that Tangled takes place three years before the main events in Frozen.

With that said, The Art of Frozen by Charles Solomon has perhaps the most official statement on the era it takes place: the 1840's.^1 Which means that depending on when Frozen takes place (Disney is never exact), Tangled could take place during either the 1830's or 1840's. It is very safe to say, as confirmed by Charles Solomon in his book, that the main portion of the influence for Frozen was the 19th century.

^1 The Art of Frozen, p. 65. This is also somewhat confirmed by Jean Gillmore who designed the costumes for Frozen. "... True, my general approach was to meld the historic silhouettes of 1840 Western Europe (give or take), with the shapes and garment relationships and details of folk costume in early Norway, circa 19th century." ^source

Just another little tidbit, there are bicycles in the film, which would not be formally introduced until the early 19th century, and would not become popular until later. This corroborates the 19th century date.

Chocolates as they are seen won't be available before the tail end of the 1700s, bicycles in the 1820s (since Anna is a child still that means you'd have to add at least 5 years). The Credits identify the foreign dignitaries as ambassadors from Germany and Ireland, the German Confederation was established in 1815, although Ireland wasn't formally a country until the 1900's I'm willing to bet that's an oversight. What we dont see are steamships, which means that it takes place before 1860 (assuming the visitors would use one, all the ships we see are sail vessels). Best guess is 1830-45

This was taken from the trivia section from Rapunzel's page on disney.wikia.com

Two official sources set the time frame in two mutually incompatible times: -According to the makers of Tangled, the film takes place in the 1780's, although the presence of books such as The Little Mermaid, and other evidence seen on-screen, implies a date closer to the 1830-40's. -Rapunzel's cameo with Flynn in Frozen, which is set in the 1840's (confirmed in The Art of Frozen), places Tangled in or around the same era. Fashion historians point that the style of clothing worn by men and women is more compatible with an early-to mid 19th century than the late 18th century.

Bonus fact! Mother Gothel (from Tangled) has a dress typical of the Renaissance period. In Germany (where Tangled takes place), the Renaissance was in the 15th and 16th centuries. Flynn states in the prologue of the movie that several centuries passed before Gothel kidnapped Rapunzel, meaning that it was more than 100 years. This puts us in the 18th and 19th centuries. As stated previously, though, Frozen was confirmed to take place in the 1840's by The Art of Frozen, meaning Gothel survived at least 300 years by singing to the flower!

I'm going to throw out a completely different theory. Frozen takes place in modern times, specifically in or after 1975.

Proof: In Elsa's song while she is building her ice castle, she sings the word "fractal" to refer to the fractal nature of the ice she was conjuring. The term "fractal" was first used by mathematician Benoît Mandelbrot in 1975.