Pyramid of broken strings

Perl, 46 + 1 = 47 bytes

Run with the -n flag

say s/.{$=}(?=.)/$&|/gr while($==y///c/++$,)-2

Try it online!

Code breakdown

-n                                              #Reads input into the $_ variable
say s/.{$=}(?=.)/$&|/gr while($==y///c/++$,)-2
                                 y///c          #Transliteration.  Implicitly operates on $_, replacing every character with itself and counting replacements
                                                #y///c effectively returns the length of $_
                                      /++$,     #Increments $, (which starts off at 0) and divides the length of $_ by $,
                              $==               #Stores the result of this division into $=
                                                #$= forces its contents to be an integer, so it truncates any decimal
                             (             )-2  #Returns 0 if $= is equal to 2
                        while                   #Evaluates its RHS as the condition.  If truthy, evaluates its LHS.
    s/          /   /gr                         #Substitution.  Implicitly operates on $_.
                                                #Searches for its first argument and replaces it with its second argument, repeating until it's done, and returns the new string.  $_ is not modified.
      .{$=}                                     #Looks for a string of $= characters...
           (?=.)                                #...that is followed by at least one non-newline character, but does not include this character in the match...
                 $&|                            #...and replaces it with itself followed by a pipe character.
say                                             #Output the result of the substitution.

Pyth, 16 bytes

Vh/lQ3j\|cQ/lQhN

V                # For N in range(1, \/ )
 h/lQ3           # 1+lenght(input)/3
      j\|        # join with '|'
         cQ      # chop input in
           /lQhN # lenght(input)/(N+1) pieces

try here

C, 145 131 128 125 bytes

l,n,i=1,j;f(char*s){l=strlen(s);puts(s);do{n=l/++i;for(j=0;j<l;)j&&(j%n||putchar('|')),putchar(s[j++]);puts("");}while(n>2);}

This is a function that takes a string as its argument and prints the output to STDOUT.

l,n,i=1,j;       // declare some variables
f(char*s){       // declare the function
l=strlen(s);     // get the length of the string
puts(s);         // output the initial version, with trailing newline
do{n=l/++i;      // n is the number of characters per "section",
                 //  and we'll do-while n>2 to stop at the right time
for(j=0;j<l;)    // loop through the characters of the string
j&&(             // if j != 0,
j%n||            // and j % n == 0,
putchar('|')),   // insert a | before this character
putchar(s[j++]); // print the character
puts("");        // print a newline after the loop
}while(n>2);}