OS/2 and Windows NT have an interesting and checkered common history. Until late 1990, the operating system eventually released as Windows NT was known as NT OS/2. Despite its name, NT had very little in common with OS/2 as it existed at the time (that is, OS/2 1.x) in terms of design or source code. The core NT design team led by Dave Cutler, mostly consisting of ex-Digital programmers, had very little experience with OS/2 or even PCs. At the same time, the plan was to provide compatibility with existing OS/2 applications and for OS/2 to be the dominant ‘personality’ of NT.
The NT kernel’s (or more correctly the NT Executive’s) design was radically different from the design of OS/2 1.x. While OS/2 1.x was a 16-bit OS designed exclusively for the segmented architecture of the Intel 286/386 CPUs, NT was a portable 32-bit OS with paged virtual memory, deliberately ported to the 386 PC platform relatively late in its development cycle. While OS/2 could not run on anything but a 286/386 without a complete rewrite, NT could not run on a 286 ever.
Dave Cutler did not get along at all with Gordon Letwin, Microsoft’s chief OS/2 architect. Cutler also had many clashes with Darryl Rubin, the lead developer of LAN Manager (which was OS/2 based at the time). 
The relationship between NT and 32-bit OS/2 is very murky. It is known that Microsoft was working on 32-bit OS/2 2.0 in the late 1980s. It is also known that NT was initially meant to support a 32-bit OS/2 API, but that plan was later scrapped for obvious political reasons. At least one feature—exception handling—is so similar between OS/2 2.0 and NT and at the same so unique in context of other operating systems that it was clearly designed by the same group of people, and the implementations were intended to be compatible.
In 1990, after Microsoft internally decided to dump OS/2 and concentrate entirely on Windows, most of Microsoft’s OS/2 developers were reassigned to NT. However, that did not influence the basic design of NT, which had been created long before then. NT adopted several technologies from OS/2, but very little or no code (16-bit Intel assembly code was of little use within NT). HPFS and FAT filesystem support was written from scratch for NT, the NDIS network driver interface underwent a major overhaul, LAN Manager components were rewritten.
On a more mundane level, OS/2 was the platform used to build NT before NT became self-supporting. After all, who wanted to develop on DOS…. Aside from editors and compilers, OS/2 also ran the Intel i860 emulator used in the very early stages of NT development, before any hardware was available.
After the Microsoft-IBM split, the relationship between NT and OS/2 became somewhat schizophrenic. Microsoft sometimes tried to pretend that OS/2 had never existed and sometimes loudly bad-mouthed anything IBM did with OS/2. At the same time, NT (up to and including Windows 2000) shipped with an OS/2 subsystem which ran character-mode 16-it OS/2 applications. Microsoft also had a Presentation Manager add-on for NT which supported OS/2 GUI applications, a semi-secret product hinted at in documentation but never actually advertised.
In the end, Windows NT survived and OS/2 did not, although OS/2’s demise had arguably more to do with Windows 3.1 and Windows 95 than NT.