Cutting it rather thin

[NDQuattro]

Hello gents.

[COLOR=maroon][FONT=Segoe UI]Ivy : [/FONT][/COLOR][COLOR=#333333][FONT=Segoe UI]Congratulations on winning the order! The deadline for the delivery is 30 days from now.
[/FONT][/COLOR][COLOR=#333333][FONT=Segoe UI][COLOR=maroon]Smith : [/COLOR]30 days? Wow! [B]That’s cutting it rather thin.[/B] I asked for a 45-day period.[/FONT][/COLOR]
[COLOR=#333333][FONT=Segoe UI][COLOR=maroon]Ivy : [/COLOR]Come on. You know we can’t afford waiting for 45 days for the material. Thirty days is the industry standard.[/FONT][/COLOR]
[COLOR=#333333][FONT=Segoe UI][COLOR=maroon]Smith : [/COLOR]You’re right Mrs. Ivy, but you’ve asked for a special design modification. That’s going to take time. I don’t want to disappoint you later.[/FONT][/COLOR]
[COLOR=#333333][FONT=Segoe UI][COLOR=maroon]Ivy : [/COLOR]Hmm.. How soon do you think you can deliver it?[/FONT][/COLOR]
[COLOR=#333333][FONT=Segoe UI][COLOR=maroon]Smith : [/COLOR]Well, I like working with you, so I am going to make an extra effort. Let’s fix it at 36 days.[/FONT][/COLOR]
[COLOR=maroon][FONT=Segoe UI]Ivy : [/FONT][/COLOR][COLOR=#333333][FONT=Segoe UI]Okay, Mr. Smith. That sounds reasonable.[/FONT][/COLOR]

What is that phrase? Cutting it rather thin? Cut what? Wood?

 

[MikeNewYork]

It refers to time, specifically less time than is needed.

 

[emsr2d2]

In BrE, we call that "cutting it fine" or "cutting it close".

 

[MikeNewYork]

We use the "close" version in AmE also.

 

[gurpreetgill4u]

I am not a teacher

I believe It means that 30 days deadline would be too short and they would barely make it in 30 days.
Cut it means "to be enough or sufficient to deal with a problem" and thin means "insufficient,barely" here.
http://www.pride-unlimited.com/probono/idioms2.html
Scroll down to cut the mustard

I am not a teacher

 

[MikeNewYork]

I have not heard the "cut it fine" version.