Well, in that particular sentence, if you used "could" you would end up with "everybody could escape" which just says that they had the ability to escape, not that they did escape.
"everyone managed to escape" or "everyone was able to escape" has the correct meaning in that people actually escaped.

The phrase "Able to manage to" really isn't used that much now days. But it basically is just a long way of saying the same thing as "able too" if you used "could" in the sentence, it would end up like this "The fire spread though the building quickly, but everyone could escape" basically saying That they had the ability to escape, not that they took action and did it.

Also, "Be able to" has a bit of a different meaning then "managed to"
Below are two sentences One is used for "Managed" and the other is used for "be able to"

1. "The fire spread through the building quickly, but everyone managed to escape."

2. "The fire is spreading through the building quickly, will the people be able to escape?"

Those are two good examples on how two use both of those phrases.

I hope I helped you some, keep up the great work sand-san. ^^v

"I was able to manage to" is a little strange. It sounds more redundant than normal sentences. The meanings of "be able to" and "manage to" are similar, but can have slight differences (I believe).

"Can manage to" can be used to say that it was difficult, but you were "able to do" it.
"Be able to" does not say either way how difficult it was.

I believe your textbook is correct that in proper formal grammar you shouldn't use the word "could" for past tenses of positive sentences. I've seen:
For just the ability do something, 'was able to' and 'could' can both be used.
For the the ability to do something and that was done, use 'was able to'.

That being said, it's very rare for people to have perfect grammar. I wouldn't be surprised to hear native speakers use "could" in either case.

Try saying:
"I finally managed to be able to (sth)."

I was able to manage to... does sound redundant.
In the right context you could say it.
"I was able to manage my finances and pay all my bills."
I believe that it is grammatically correct, but not used much.
"able" is an adjective and "manage" is a verb.
I think it's the double verb that sounds odd.
"Manage to fix" has 2 verbs,
"Manage my finances" has a verb and a noun.
I better stop there, I'm bordering on rambling now..haha

I think you can use "could" in that sentence but it doesn't sound that natural. I think these all mean the same:

The fire spread through the building quickly, but everybody was able to escape.
The fire spread through the building quickly, but everybody managed to escape.
The fire spread through the building quickly, but everybody could escape. (Sounds a little awkward)

In this context "be able to" and "manage to" means the same.

"able to manage to" isn't correct. You can say "able to manage" (管理することができる） but it has a different meaning.

Examples of "able to":
http://jisho.org/sentences?jap=&eng=able+to

Examples of "managed to":
http://jisho.org/sentences?jap=&eng=managed+to

Also "managed" and "managed to" can mean different things:
http://jisho.org/sentences?jap=&eng=managed

Hello, Sand.
Your English skills are really amazing!!

You have less and less mistakes with every journal ~ :D
~Amazing!~

Everyone has already corrected your diary, so I can't add anything! ^_^ Great job Sand!

I think "I was able to manage to..." is okay, but "I was able to somehow..." or "I was somehow able to..." sounds a bit more natural to me.

> madao,

"Could" in this case means having ability to escape. I see.
Thank you for your explanation!

> ハリマ☆ハリオ,

Thank you for your explanation about the difference.
By the way, you changed your nickname, didn't you? (^.^)

> lailyx,

I see. "I was able to manage to" sounds redundant.
Thank you for your explanation!

> 三 lines,

Oh, "I finally managed to be able to (sth)." I see.
Thank you for your corrections and comment!

> EroOyaji,

Double verb sounds odd. I see.
Thank you for your explanation!

> Tip,

The information you gave is very helpful.
Thank you for your explanation!

> Kame Sennin,

Oh, really? I'm glad to hear that! (^.^)
Thank you for your comment!

> ｼﾞｬｽ,

Thank you for your comment!

> cuavsfan,

Oh, using "somehow." I see.
Thank you for your explanation!

The difference between "able to" and "manage to" is that "manage to" implies that there is some sort of struggle involved. "Able to" just describes what happened, without letting the reader know whether it was difficult or easy.

I managed to start the car.--> I started the car, but it was troublesome.
I was able to start the car.-->I started the car. It may have been troublesome.

Shin,

Oh, there is such a difference between "manage to" and "was able to." I see.
Thank you for your corrections and explanation!