If your code can accommodate the error (whatever it is) that's occurring early on, this can be a reasonable way of doing it, but I'd always have to look twice at it in a code review because it's fairly unusual for the code to be able to just ignore errors like that. The code is roughly equivalent to:

try {
    //my code   
} catch (e) {
}

try {
    //my code
} catch(e) {
}

try {
    //my code   
} catch(e) {
}

...but using promises instead. So it's a bit suspect, but can be correct, for the same reasons the above is a bit suspect but can be correct if you need to do a series of things, one at a time, and have each of them done even if the previous one fails.

Beware that it means app in the subsequent fulfillment handler will be undefined:

.then(function(response) {
    //my code   
})
.catch(e => {})

.then(function(app) { // <=== `app` is `undefined` here
    //my code
})
.catch(e => {})

.then(function() {
    //my code   
})
.catch(e => {})

Answer posted before, is correct.

I just want to insert my 2 cents.

You can have some fun with one helper function (or use something more fancy like whole Either monad thingy from fp-ts package)

const run = async (fn) => {
    try {
       const result = await fn()
       return [result, null]
    } catch (err) {
        return [null, err]
    }
}

and write code without try\catch or .then\.catch

const [response, error] = await run(() => fetch('asdfasdf'))
const app = buildApp(response.ok ? await response.json() : { fallback: 'data' })

const [appResponse, appError] = await run(async () => {
  await app.compileTemplate()
  return app.buildResponse()
})

if (appResponse) {
 // ...
}

Or more useless approach, you can throw a custom error, so your first .catch block will be able to do something.

class AppFromResponseError extends Error {
  constructor(message, fallbackApp) {
    super(message)
    this.name = "ResponseError"
    this.app = "fallbackApp"
  }
}

builder
  .then(function(response) {
    if (response.ok !== true) 
      throw new AppFromResponseError('not ok', minimalApp)
  })
  .catch(e => {
    if (e.app) return e.app
  })
  .then(app => { /* some kind of app will be here */})