It has been stated that the catch rate does increase when you get a nice, great or excellent throw.

throw bonuses will confer a boost to the base catch rate for that Pokemon for that throw and all future throws.

Forbes and many other sources declare that this is true as-well.

The longer you hold onto your Pokéball, the smaller this ring gets before resetting to its largest size. You get an increased capture rate for hitting inside this ring

Updated Answer: The Silph Road's detailed analysis concludes that throw bonus does increase your catch rate.* https://thesilphroad.com/science/pokemon-go-throw-bonuses

See @vemonus answer down below: https://gaming.stackexchange.com/a/288122/75659

Answer: No one officially knows yet. It's frequently a controversial topic in terms of anecdotal evidence: some players say they notice an increase in capture rate, others don't - however, in general, most players seem to be in agreement that it does confer a catch rate bonus.

Below is an explanation of why the capture rate bonuses detailed on certain websites (such as ones in the above answers) are misleading and based on unverified claims.

The closest thing we had to an indication that it does affect the rate (besides anecdotal evidence) is within the protobuf file for the game. This is a list of numbers within the game that detail things like medal achievement requirements, experience bonuses, gym experience numbers, etc. I've provided the link below:

https://gist.github.com/anonymous/077d6dea82d58b8febde54ae9729b1bf

Within this protobuf we can see that there are only two values related to throw types. The first is an item detail for each type of ball, e.g. for greatball:

Items {
  TemplateId: "ITEM_GREAT_BALL"
  Item {
    UniqueId: ITEM_GREAT_BALL
    ItemType: ITEM_TYPE_POKEBALL
    Category: ITEM_CATEGORY_POKEBALL
  }
}

and the second lists "thresholds" for each type of throw:

Items {
  TemplateId: "ENCOUNTER_SETTINGS"
  EncounterSettings {
    SpinBonusThreshold: 0.5
    ExcellentThrowThreshold: 1.7
    GreatThrowThreshold: 1.3
    NiceThrowThreshold: 1
    MilestoneThreshold: 100
  }

Some people saw these numbers and assumed it meant the catch rate multiplier...however others speculate that:

1. this doesn't make sense for spin bonus threshold, since it'd cut the rate in half. it makes more sense if the spin threshold meant (as its name indicates) how much spin is required before its counted as a spin throw
2. the word threshold doesn't make sense given the context (you'd think something more like GreatThrowMultiplier would be more appropriate) and
3. these numbers more accurately line up with (1 - radius) of the size of the inner circle upon impact.

To elaborate on point 3:

The answer here says for nice, great, and excellent the percentage of the inner circle to outer circle diameter required would respectively be: 100%, 70%, and 30%.*

To clarify/reiterate for people unfamiliar with throw types: you have to not only have a smaller circle to get a bonus, you have to hit the pokeball inside the circle as well.

To translate these numbers into the threshold numbers, all you have to do is see that:

let x = the value upon impact based on the distance

• x = 1 means your pokeball hit the pokemon while the inner circle was the same size as the outer circle
• x = 2 means the inner circle had a radius of 0

Take (2 - x) and you see that the thresholds as detailed in the protobuf line up with the percentages.

For example:

2 - ExcellentThrowThreshold
            =
          2 - 1.7
            =
           0.3
            =
           30%

Here is a link that details actual data communicated with the servers and basically confirms that the values of 1.00 - 1.99 were the thresholds when that protobuf was published (and also explains how these numbers have to do with why there was no bonus exp for throw types for one patch: the 'x' value was changed from 1.00 - 1.99 to 0.00 - 0.99 and thus no one could achieve the 1.3/1.7 required for the bonus except for hackers).

*NOTE: This is not a definitive source, but it lines up so perfectly with the explanation and numbers that it's very likely true.

This was posted today on the Silph Road subreddit and has pretty conclusive evidence that catch rate is indeed influenced by throw bonuses.

The controls of the experiment were as follows:

1. Only samples collected before the recent TYPE CATCH BONUS mechanic launched were studied
2. This dataset only includes balls that were thrown STRAIGHT (See Note 1)
3. This study was only conducted on POKE BALLS
4. This study excluded Pokemon caught via INCENSE or LURE PATCHES
5. Only Pokemon whose BASE CATCH RATE is .4 were included (e.g. Rattata, Pidgey, etc)
6. We used % OF MAX CP per species as a basic way to estimate Pokemon level (See Note 2)

Here is a table with the results:

From these results, it looks like the 1.1x Nice, 1.3x Great, and 1.5x Excellent multipliers are a bit exaggerated.

From the numbers above, it looks like the actual bonuses are roughly:

• Nice: 1.07x
• Great: 1.20x
• Excellent: 1.31x

Additionally, another article by the Silph Road was posted yesterday that confirms that curveballs also increase catch rate. For the full details of this experiment, look here.

Here is a chart containing the results of the experiment.