Cfrgtkky

The following code:

documentclass[tikz, border=1mm]{standalone}



usepackage{tikz}

usetikzlibrary{backgrounds, positioning, fit}

begin{document}



newcommand{boxdist}{3cm}

newcommand{textwdh}{2cm}



begin{tikzpicture}[

    box/.style={rectangle, rounded corners, draw=black, text width=15mm},

    ]

    node (A) [text width=textwdh] 

    {Blaaaaaa bbbbbbbb newline Some text text text text};

    node (B) [right= boxdist of A.west, text width=textwdh] {More text text text};

    node (C) [right= boxdist of B.west, text width=textwdh] {Even more text test};

    node (D) [right= boxdist of C.west, text width=textwdh] {And even more text};



    begin{scope}[on background layer]

        node [box, inner sep=11mm, fill=black!40, fit={(A) (B) (C) (D)}] {};

    end{scope}



    begin{scope}[on background layer]

        node [box, inner sep=9mm, fill=black!30, fit={(B) (C) (D)}] {};

    end{scope}



    begin{scope}[on background layer]

        node [box, inner sep=7mm, fill=black!20, fit={(C) (D)}] {};

    end{scope}



    begin{scope}[on background layer]

        node [box, inner sep=5mm, fill=black!10, fit={(D)}] {};

    end{scope}



end{tikzpicture}



end{document}

produces the following output:

I want the text in each box to start in the upper left corner. I also want the text to have the same width in all boxes. How do I do this?

Like this?

documentclass[tikz, border=3mm]{standalone}

usetikzlibrary{backgrounds, fit, positioning}



begin{document}

    begin{tikzpicture}[

  node distance = 4pt and 4pt,

     box/.style = {rectangle, inner sep=0pt, text width=21mm},

FIT/.style args = {#1/#2}{rectangle, rounded corners, draw, semithick,

                    fill=gray!#1, inner sep=2pt, fit=#2,

                    node contents={}}

                        ]

node (A) [box] {Blaaaaaa bbbbbbbb Some text text text text};

node (B) [box, below right=of A.north east] {More text text text};

node (C) [box, below right=of B.north east] {Even more text test};

node (D) [box, below right=of C.north east] {And even more text\~};

%

begin{scope}[on background layer]

node (d) [FIT=0/(D)];

node (c) [FIT=0/(C) (d)];

node (b) [FIT=0/(B) (c)];

node (a) [FIT=0/(A) (b)];

%

node [FIT=60/(a)];

node [FIT=40/(b)];

node [FIT=20/(c)];

node [FIT=10/(d)];

end{scope}

    end{tikzpicture}

end{document}

addendum:
A case when the text in nodes are vertical is centered (as you request in comment):

documentclass[tikz, border=3mm]{standalone}

usetikzlibrary{backgrounds, fit, positioning}



begin{document}

    begin{tikzpicture}[

  node distance = 4pt and 4pt,

     box/.style = {rectangle, inner sep=0pt, text width=21mm},

FIT/.style args = {#1/#2}{rectangle, rounded corners, draw, semithick,

                    fill=gray!#1, inner sep=2pt, fit=#2,

                    node contents={}}

                        ]

node (A) [box] {Blaaaaaa bbbbbbbb Some text text text text};

node (B) [box, right=of A] {More text text text};

node (C) [box, right=of B] {Even more text test};

node (D) [box, right=of C] {And even more text};

%

begin{scope}[on background layer]

node (d) [FIT=0/(D)];

node (c) [FIT=0/(C) (d)];

node (b) [FIT=0/(B) (c)];

node (a) [FIT=0/(A) (b)];

%

node [FIT=60/(a)];

node [FIT=40/(b)];

node [FIT=20/(c)];

node [FIT=10/(d)];

end{scope}

    end{tikzpicture}

end{document}