なくて

なくて form “Not”
い Adjective くない form →くなくて “Not __ and 

Creating this negative て form is pretty straightforward. The ない form  is (treated) as an い adjective, and then conjugated to the い adjective て form (くて）

With い Adjectives

Theなくて form here functions as expected, as a adjective of the opposite meaning’s て form.
Adjectives must still connect to adjectives.

Examples

• やさしくなくて、ひどい人だ – (A) not nice and horrible person.
• 楽しくなくて、つまらない – Not fun and boring. 

With Nouns and な Adjectives

なくてform
Noun/な adjective + じゃなくて “not a”

Nouns and な Adjectives use じゃない (is not), conjugated to じゃなくてwhich is a state. This changes the meaning slightly to “not a__, but a___”

Examples

• 猫じゃなくて、犬だ。-Not a cat, (a) dog.
• それは　えんぴつじゃなくて、ペンだ – That’s a pen, not a pencil.
• しずかじゃなくて、うるさいよるだった – Not a quiet, but a loud night.

With Verbs

Verb なくて (Didn’t, so…)
Verb ない form →なくて

However, when used with a verb, the meaning changes to “didn’t do X, so..“. This must show cause and effect.
AなくてB (Didn’t A, so B).

Absent other grammar points like “if” or もいい, verbs in this form tend to produce past tense or present progressive sentences. 

Examples

• 勉強しなくて、不合格だった – (someone) Didn’t study, so failed.
• 告白できなくて、チャンスをうしなった。ーDidn’t confess so lost their chance.
• 朝ごはんを食べなくて、おなかがすいた。- Didn’t eat so (is) hungry.
• 遊ばなくて落ち込んでいる – Didn’t play so (is) now depressed.

See also: ないで

Comparison Table