Pure bash: 97

s=$1;shift;for((l=${#s}/$#,m=${#s}-l*$#,i=1;i<=$#;p+=q,i++)){
printf "${s:p:q=i>m?l:l+1}">${!i};}

or with exactly same length:

s=$1;shift;for((a=${#s},l=a/$#,m=a-l*$#,i=1;i<=$#;p+=q,i++)){
printf "${s:p:q=i>m?l:1+l}">${!i};}

As a function: (p= is only required for second run)

dog() { p=
    s=$1;shift;for((l=${#s}/$#,m=${#s}-l*$#,i=1;i<=$#;p+=q,i++)){
    printf "${s:p:q=i>m?l:l+1}">${!i};}
}

Tests

$> rm *
$> dog "Dogs vs Cats" a.txt b.txt c.txt
$> ls -g
total 12
-rw-r--r-- 1 user 4 May 13 22:09 a.txt
-rw-r--r-- 1 user 4 May 13 22:09 b.txt
-rw-r--r-- 1 user 4 May 13 22:09 c.txt
$> cat {a,b,c}.txt;echo
Dogs vs Cats
$> 

All files is 4 byte len and concatenated in right order, contain "Dogs vs Cats".

$> rm *
$> dog "$(printf "%s" {0..9} {a..c})" {a..e}.txt 
$> ls -g
total 20
-rw-r--r-- 1 user 3 May 13 22:09 a.txt
-rw-r--r-- 1 user 3 May 13 22:09 b.txt
-rw-r--r-- 1 user 3 May 13 22:09 c.txt
-rw-r--r-- 1 user 2 May 13 22:09 d.txt
-rw-r--r-- 1 user 2 May 13 22:09 e.txt
$> cat *;echo
0123456789abc
$> 

Firsts files is 3 byte len and last only 2, concatenated by alphabetic order, contain "0123456789abc".

Explanation (ungolfing):

If you hit: declare -f dog, bash will answer:

$> declare -f dog
dog () 
{ 
    p=;
    s=$1;
    shift;
    for ((l=${#s}/$#,m=${#s}-l*$#,i=1; i<=$#; p+=q,i++))
    do
        printf "${s:p:q=i>m?l:l+1}" > ${!i};
    done
}

This could be written:

dog2() {
    local string
    local -i i oneMore partLen partQuant position
    string=$1
    shift
    partLen=" ${#string}/$# "
    oneMore=" ${#string}-partLen*$# "
    for (( i=1; i<=$#; i++ )) {
        if (( i <= oneMore )); then
            partQuant=" partLen + 1 "
        else
            partQuant=partLen
        fi
        printf "${string:position:partQuant}" > ${!i};
        position+=partQuant
    }
}

Note: by using local -i, I could avoid $(( .. ))!

Ruby, 93 87 bytes

Full program using command line arguments.

If I could use s.slice! to mutate the string, I'd do that instead of needing to use s[c..-1], but Ruby doesn't let you mutate the strings from argv without duplicating them first

s,*t=$*
d,r=s.size.divmod t.size
t.map{|e|open(e,?w)<<s[0,c=(0>r-=1)?d:d+1];s=s[c..-1]}

PHP, 107 bytes

The golfed code:

for($i=1;++$i<$argc;fputs(fopen($argv[$i],w),substr($s=$argv[1],($i-2)*$l=ceil(strlen($s)/($argc-2)),$l)));

The detailed code:

$len = ceil(strlen($argv[1])/($argc - 2));
for ($i = 2; $i < $argc; $i ++) {
    $fh = fopen($argv[$i], 'w');
    fputs($fh, substr($argv[1], ($i - 2) * $len, $len));
    fclose($fh);          // omitted in the golfed version
}

Python - 181 bytes

import sys
a=sys.argv
l=len
d=a[2:]
s=a[1]
n,r=divmod(l(s),l(d))
p=0
for i in range(l(d)):
    with open(d[i],'w') as f:
        o=n+int(i<=n)
        f.write(s[p:p+o])
        p+=o

Pyth - 12 bytes

.wMC,cl.zz.z

Uses builtin split function and then uses splat-map on the write function. Doesn't work online.