How fast does the limit break meter increase?

Taken from FF7 Party Mechanics

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3.2 The Limit Bar

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The Limit Bar dictates when you can use a certain Limit Break. Depending on your Status Attributes, and which Limit Level you have chosen to build, it will grow quickly or slowly based on the damage you take.

A character's Limit Bar is measured in Units. Once a Limit Bar contains 255 Units, the Limit Bar is full, and the character may use their Limit Break.

The conversion rate between damage points and Limit Bar Units is as follows:

Units gained = [[300 * HP Lost / Max HP] * 256 * Status Factor / LNum]

In the above formula, Status Factor is equal to 2.0 while in Fury Status, 0.5 in Sadness Status, and 1.0 otherwise.

LNum, on the other hand, is a special number that varies for each character at each Limit Level. The following table shows the LNum value for all nine characters at each of their Limit Levels.

                   Level 1       Level 2       Level 3       Level 4
     Cloud           140           324           435           506
     Barret          129           240           374           450
     Tifa            120           334           509           566
     Aeris           200           284           322           436
     Red XIII        195           300           389           486
     Yuffie          200           300           400           480
     Cait Sith       160           180          9590 *       12760 *
     Vincent         202           244           319           426
     Cid             200           280           380           480

 * Cait Sith does not have either a L3 or L4 Limit Break - these numbers
   are still in the data, however.

The TFergusson guides for FF7 are some of the most detailed in-depth FAQ's I have ever seen. He even has a guide that has the AI script for every enemy in the game

Using the data in Scott Chamberlain's answer, I came up with a chart to represent this all in kind of a more intuitive way.

The Y axis represents how much damage you have to take, as a percentage of your maximum HP, in order to be able to use a limit break. For example, to use a level 4 limit break, Cloud must take approximately 170% of his maximum HP in damage.

This assumes that you do not have the Sadness of Fury status afflictions. To account for Sadness, double these numbers; for Fury, halve them.