The expression に限（かぎ）って in Japanese is a phrase that has three primary uses, the first is to show that (B) is an unbelievable thing about (A), the second is to express that something is limited to (A), and the third highlights that in most cases for (A), (B). Depending on the particular nuance that is being used, common translations include ‘particularly when (A), (B)’, ‘those who (A), (B)’, and ‘only when (A), (B)’.
This structure is comprised of に, and the て-form of the う-Verb 限（かぎ）る ‘to limit’. Let’s look at a few examples of each potential meaning.
First, when something about (A) is unbelievable due to what is either expected or previously known.

• うちの子（こ）に限（かぎ）って、他（ほか）の子（こ）に手（て）を出（だ）すなんて考（かんが）えられない。
  I can’t imagine that my child, in particular, would hit another kid.
• 彼（かれ）に限（かぎ）って、こんな簡単（かんたん）なことでミスをする訳（わけ）が無（な）い。
  There is no way that he, in particular, would make a mistake on something this simple.

Second, when something is limited to (A).

• 武（たけし）くんと釣（つ）りに行（い）く日（ひ）に限（かぎ）って、いつも雨（あめ）が降（ふ）るんだよな。
  Only on days when I go fishing with Takeshi-kun does it always rain.
• 先輩（せんぱい）は忙（いそが）しい日（ひ）に限（かぎ）って、飲（の）みに誘（さそ）ってくる。
  My senpai invites me to go drinking with him only on days that I am busy.

Third, when (B) is true in most cases concerning (A).

• 「クレームを入（い）れるぞ」って言（い）う人（ひと）に限（かぎ）ってクレームを入（い）れないからあまり気（き）にしていない。
  I’m not too worried because only those who say, ‘I'm going to file a claim’, never file a claim.
• 「絶対（ぜったい）後（あと）でやる」って言（い）う人（ひと）に限（かぎ）って、いつまでもやらない人（ひと）が多（おお）い。
  Only those who say, ‘I'll definitely do it later’, don't do it at all.

While each of these uses can roughly be translated as ‘only when’, it will be important to keep all three meanings in mind when interpreting sentences that use に限（かぎ）って.