This paper briefly summarises the characteristics and relevant properties of nanoparticles and 32 elements which may be contained in the nanoparticles. The Instrumental Neutron Activation Analysis (INAA) is then briefly described, because this method enables determination of elements, including those with extremely low weights. Element characteristics are examined and the minimum determinable weights of 12 elements are presented at a neutron flux density of 3·1014 n/(cm2 ·s) established in a nuclear reactor. The gold and silver nanoparticles have especially high significance. While Au, In and Ir weights can be determined in 1 fg (1·10-15 g), order of magnitude (= 1·10-15 g), As, Co, Ga, La and Na are in 10 fg and Ag, Cd and Hg are determined in 100 fg. Experimental verification of these threshold values can reveal if these values could be even lower for some elements. Finally, the results presented herein can help evaluate the procedures used for monitoring nanoparticles in organisms and in the living environment.