You have too many basis functions and therefore you end up with strongly correlated fit coefficients.
I'll try to construct an example to show what is happening.
Let's assemble some demo data:
data = Table[{x, x^4/100 + 10^-6 RandomVariate[NormalDistribution[]]},
{x, 0, 1/5, 1/2000}];
ListPlot[data]

Now fit these data with the basis functions you provide:
p = {5/6, 1, 5/3, 11/6, 2, 5/2, 8/3, 17/6, 3, 19/6, 10/3, 7/2, 11/3, 23/6, 4};
fit = LinearModelFit[data, x^p, x, IncludeConstantBasis -> False];
fit["ParameterTable"]
$$
\begin{array}{l|llll}
\text{} & \text{Estimate} & \text{Standard Error} & \text{t-Statistic} & \text{P-Value} \\
\hline
x^{5/6} & -1.27552 & 3.40025 & -0.375125 & 0.707774 \\
x & 9.61197 & 25.1332 & 0.382441 & 0.702345 \\
x^{5/3} & -19037.2 & 47842.1 & -0.397918 & 0.69091 \\
x^{11/6} & 169800. & 425903. & 0.398681 & 0.690349 \\
x^2 & -496734. & 1.24674\times 10^6 & -0.398425 & 0.690537 \\
x^{5/2} & 3.242\times 10^7 & 8.25662\times 10^7 & 0.392655 & 0.694791 \\
x^{8/3} & -2.1707\times 10^8 & 5.57458\times 10^8 & -0.389393 & 0.6972 \\
x^{17/6} & 7.45553\times 10^8 & 1.93348\times 10^9 & 0.385603 & 0.700003 \\
x^3 & -1.64553\times 10^9 & 4.31499\times 10^9 & -0.381353 & 0.703151 \\
x^{19/6} & 2.50338\times 10^9 & 6.64542\times 10^9 & 0.376707 & 0.706598 \\
x^{10/3} & -2.67567\times 10^9 & 7.19809\times 10^9 & -0.371719 & 0.710306 \\
x^{7/2} & 1.98576\times 10^9 & 5.41912\times 10^9 & 0.366436 & 0.714241 \\
x^{11/3} & -9.78551\times 10^8 & 2.71142\times 10^9 & -0.360899 & 0.718372 \\
x^{23/6} & 2.88751\times 10^8 & 8.13048\times 10^8 & 0.355146 & 0.722674 \\
x^4 & -3.86909\times 10^7 & 1.10796\times 10^8 & -0.349207 & 0.727124 \\
\end{array}
$$
Here already we see that the Standard Errors of the coefficients are very large, larger than the coefficients themselves. The problem becomes even more apparent when we check how these fit coefficients are correlated:
fit["CorrelationMatrix"] // MatrixForm
$$
\left(
\begin{array}{ccccccccccccccc}
1. & -0.999719 & 0.993488 & -0.990828 & 0.9878 & -0.976859 & 0.972701 & -0.968338 & 0.963795 & -0.959096 & 0.954265 & -0.949322 &
0.944285 & -0.939171 & 0.933995 \\
-0.999719 & 1. & -0.995896 & 0.993729 & -0.991176 & 0.981551 & -0.977795 & 0.973815 & -0.969638 & 0.965289 & -0.96079 & 0.956162 &
-0.951425 & 0.946597 & -0.941692 \\
0.993488 & -0.995896 & 1. & -0.999768 & 0.999093 & -0.994731 & 0.992602 & -0.990185 & 0.987507 & -0.984593 & 0.981469 & -0.978156 &
0.974675 & -0.971046 & 0.967287 \\
-0.990828 & 0.993729 & -0.999768 & 1. & -0.999778 & 0.996701 & -0.994974 & 0.992945 & -0.99064 & 0.988087 & -0.985309 & 0.982329 &
-0.979168 & 0.975846 & -0.972381 \\
0.9878 & -0.991176 & 0.999093 & -0.999778 & 1. & -0.998186 & 0.996856 & -0.99521 & 0.993276 & -0.991081 & 0.988647 & -0.985999 &
0.983157 & -0.980142 & 0.976971 \\
-0.976859 & 0.981551 & -0.994731 & 0.996701 & -0.998186 & 1. & -0.999817 & 0.999286 & -0.998433 & 0.997285 & -0.995864 & 0.994193 &
-0.992296 & 0.99019 & -0.987897 \\
0.972701 & -0.977795 & 0.992602 & -0.994974 & 0.996856 & -0.999817 & 1. & -0.999826 & 0.99932 & -0.998509 & 0.997415 & -0.996062 &
0.994472 & -0.992664 & 0.990657 \\
-0.968338 & 0.973815 & -0.990185 & 0.992945 & -0.99521 & 0.999286 & -0.999826 & 1. & -0.999834 & 0.999353 & -0.998581 & 0.99754 &
-0.996252 & 0.994736 & -0.993013 \\
0.963795 & -0.969638 & 0.987507 & -0.99064 & 0.993276 & -0.998433 & 0.99932 & -0.999834 & 1. & -0.999842 & 0.999384 & -0.998649 &
0.997658 & -0.996431 & 0.994987 \\
-0.959096 & 0.965289 & -0.984593 & 0.988087 & -0.991081 & 0.997285 & -0.998509 & 0.999353 & -0.999842 & 1. & -0.99985 & 0.999414 &
-0.998714 & 0.99777 & -0.996601 \\
0.954265 & -0.96079 & 0.981469 & -0.985309 & 0.988647 & -0.995864 & 0.997415 & -0.998581 & 0.999384 & -0.99985 & 1. & -0.999857 &
0.999442 & -0.998776 & 0.997876 \\
-0.949322 & 0.956162 & -0.978156 & 0.982329 & -0.985999 & 0.994193 & -0.996062 & 0.99754 & -0.998649 & 0.999414 & -0.999857 & 1. &
-0.999864 & 0.999469 & -0.998834 \\
0.944285 & -0.951425 & 0.974675 & -0.979168 & 0.983157 & -0.992296 & 0.994472 & -0.996252 & 0.997658 & -0.998714 & 0.999442 &
-0.999864 & 1. & -0.99987 & 0.999494 \\
-0.939171 & 0.946597 & -0.971046 & 0.975846 & -0.980142 & 0.99019 & -0.992664 & 0.994736 & -0.996431 & 0.99777 & -0.998776 & 0.999469
& -0.99987 & 1. & -0.999876 \\
0.933995 & -0.941692 & 0.967287 & -0.972381 & 0.976971 & -0.987897 & 0.990657 & -0.993013 & 0.994987 & -0.996601 & 0.997876 &
-0.998834 & 0.999494 & -0.999876 & 1. \\
\end{array}
\right)
$$
Some coefficients, for example #1 and #3, are almost 100% correlated!