I tried your function and numerical precision and accuracy limits were exceeded. You can alleviate this by raising the same (see Help) or using base numbers with higher accuracy and precision (more digits) or by using arbitrary precision numbers. (it's suggested you always use arbitrary precision at all times unless limited accuracy and precision, including error of calculations, are to be calculated intentionally). Next note I zero'ed some of the last digits of "a+b". This is due to calculation errors that Mathematica tracks on each +, -, /, *.
In[404]:= NMaximize[{3.6027769090403445 10^(-52) a^3 +
9.73306257869222 10^(-52) a^2 b + -3.1793546987662064 10^(-67) a^2 \
+ 6.968194393839056 10^(-52) a b^2,
a >= 0 && b >= 0 && a + b == 290015693412189000000000}, {a, b}]
Out[404]= {8.78824*10^18, {a -> 2.90016*10^23, b -> 0.}}
In[401]:= Maximize[{Rationalize[3.6027769090403445 10^(-52),
10^(-52)] a^3 +
Rationalize[9.73306257869222 10^(-52), 10^(-52)] a^2 b +
Rationalize[-3.1793546987662064 10^(-67), 10^(-67)] a^2 +
Rationalize[6.968194393839056 10^(-52), 10^(-52)] a b^2,
a >= 0 && b >= 0 && a + b == 290015693412189845814109}, {a, b}]
Out[401]= \
{415916158542085055402727940040315199561430548561835112571162290342123\
8795019140734549704991222017851010872571220634013/
47326449583022447216549007278772095603881176748731854250575980000411\
7209146543199805755858639388672, {a -> 290015693412189845814109,
b -> 0}}
I hope that helps you search where in Help you can look to resolve your question. And yes the N in NMaximize does mean calculations will have a limited width, ie, FindRoot[x^2 - 2, {x, 1}, WorkingPrecision -> 60]. Ultimately when you use "numeric approximation" functions that are built to take advantage of built-in math in your CPU, they will be limited and fail, and to get around this limit you must forego math-chip acceleration. The subject of rounding error correctly during programs is an involved one - it's suggested you use arbitrary precision where possible except when printing final results in decimal digits.
