String concatenation performance with +=

[Sammo]

String concatenation has been optimized since 2.3, so using += should
be fairly fast.

In my first test, I tried concatentating a 4096 byte string 1000 times
in the following code, and the result was indeed very fast (12.352 ms
on my machine).

import time
t = time.time()
mydata = ""
moredata = "A"*4096
for i in range(1000):
    mydata += moredata # 12.352 ms
print "%0.3f ms"%(1000*(time.time() - t))

However, I got a different result in my second test, which is
implemented in a class with a feed() method. This test took 4653.522
ms on my machine, which is 350x slower than the previous test!

class StringConcatTest:
    def __init__(self):
        self.mydata = ""

    def feed(self, moredata):
        self.mydata += moredata # 4653.522 ms

test = StringConcatTest()
t = time.time()
for i in range(1000):
    test.feed(moredata)
print "%0.3f ms"%(1000*(time.time() - t))

Note that I need to do something to mydata INSIDE the loop, so please
don't tell me to append moredata to a list and then use "".join after
the loop.

Why is the second test so much slower?