Did Eleanor Roosevelt say the quote about our deepest fear is not that we are inadequate?

No, Eleanor Roosevelt neither wrote nor said the quotation to which you refer.

The quotation (below) was written by Marianne Williamson. It is part of her poem titled, "Our Deepest Fear," which appears as prose (Chapter 7) in her book titled, "A Return to Love: Reflections on the Principles of ‘A Course in Miracles’" (1992).

Her words are commonly misattributed to Nelson Mandela, more specifically misattributed as appearing in his Inauguration Addresses delivered in Cape Town and Pretoria, South Africa, in May of 1994.

I should add that I have not seen the quotation to which you refer attributed to Eleanor Roosevelt; but I have no doubt that it has.

_"Our Deepest Fear" by Marianne Williamson_

"Our deepest fear is not that we are inadequate. /
Our deepest fear /
is that we are powerful beyond measure. /
It is our light, not our darkness, /
that most frightens us. /
We ask ourselves, who am I to be brilliant, gorgeous, /
talented and fabulous? /
Actually who are we not to be? /
You are a child of God. /
Your playing small doesn’t serve the world. /
There is nothing enlightened about shrinking /
so that other people /
won’t feel insecure around you. /
We are all meant to shine as children do. /
We were born to make manifest /
the glory of God that is within us. /
It’s not just in some of us; it’s in everyone. /
And when we let our own light shine, /
we unconsciously give other people /
permission to do the same. /
As we are liberated from our own fear, /
our presence automatically liberates others."
[ http://www.marianne.com/index.php ]

EDIT NOTE: MR. HUIE HAS CORRECTLY NOTED IN HIS ANSWER THAT THE LINES BELOW DO NOT APPEAR IN NELSON MANDELA’S MAY 1994 INAGURATION ADDRESSES. AS HE APPROPRIATELY NOTES, ATTRIBUTION OF THE REMARKS IN THE PARAGRAPH BELOW TO NELSON MANDELA IS AN URBAN LEGEND. PLEASE IGNORE THE TWO (2) SOURCES LISTED IMMEDIATELY AFTER THE PARAGRAPH BELOW.

"…Our deepest fear is not that we are inadequate. Our deepest fear is that we are powerful beyond measure. It is our light, not our darkness that most frightens us. We ask ourselves, ‘Who am I to be brilliant, gorgeous, talented… fabulous?’ Actually, who are you not to be? You are a child of God. Your playing small doesn’t serve the world. There is nothing enlightened about shrinking so that people won’t feel insecure around you. We were born to make manifest the glory of God that is within us. It’s not just in some of us; it’s in everyone.And when we let our own light shine, we unconsciously give other people permission to do the same. As we are liberated from our own fear our presence automatically liberates others…."
[ http://www.empoweredtransformations.com/quotes.html ]
[ http://www.freechild.org/quotations.htm ]

For additional information, please see:
http://www.appleseeds.org/Deepest-Fear.htm
http://skdesigns.com/internet/articles/quotes/williamson/our_deepest_fear/
http://www.hidden-wealth-keys.com/our-deepest-fear.html
http://www.quotegarden.com/confidence.html

EDIT NOTE: MR. HUIE, THANK YOU FOR THE CORRECTION. I DO NOT WISH TO ADD TO THE "URBAN LEGEND" THAT WOULD HAVE US BELIEVE THAT MRS. WILLIAMSON’S WORDS WERE PART OF NELSON MANDELA’S INAUGURAL ADDRESSES.