Is it a wordinian?

Python, 52 bytes

lambda w,d:len({len(x)for x in d if x in w})==len(w)

An anonymous function that takes a word w and dictionary d. It takes the words in d that are substrings of w, makes a set of their lengths, and then checks that there as many distinct lengths as there are letters in w.

Python 3, 108 bytes

lambda w,d,r=range:all(any(i in d for i in j)for j in[[w[i:i+s]for i in r(len(w)+1-s)]for s in r(1,len(w))])

An anonymous function that takes input, via argument, of a word w as a string and a dictionary d as a list of strings and returns True or False.

How it works

The first step is a list comprehension that generates a list of lists of all substrings of w excluding w, grouped by length. For example, for 'stage', the list [['s', 't', 'a', 'g', 'e'], ['st', 'ta', 'ag', 'ge'], ['sta', 'tag', 'age'], ['stag', 'tage']] is generated. This is achieved by looping over all valid start indices i for each substring length s, and slicing every s-length substring using w[i:i+s]. For each list in this list, the presence of each substring in the dictionary is checked; calling any returns a hit if at least one match for a given length is found. Finally, calling all checks if a match has been found for all substring lengths, and the result of this is returned.

Try it on Ideone

Ruby, 44 bytes

• 7 Bytes off thanks to @NotThatCharles and his set operator tricks!
• 2 bytes off thanks to @Jordan with the Ruby 2.3 safe navigation operator trick w[x]&.size :)

->w,d{[*1..w.size]-d.map{|x|w[x]&.size}==[]}

It's an anonymous functions which takes a word w and a dictionary (array of words) d. Creates two arrays: The first containing the numbers 1 up to and including the length of w; The second array is d with each word mapped to their size if they are a substring of w, otherwise nil. Then it does set substraction to check whether the second array contains all element of the first array.