Conditions of double-stranded DNA precipitation by the polyamines spermidine and spermine have been determined experimentally and compared to theoretical predictions. The influence of the concentrations of DNA and added monovalent salt, and of the DNA length has been investigated in a systematic manner. Three regimes of DNA concentrations are observed. We clarify the dependence of these regimes on the monovalent salt concentration and on the DNA length. Our observations make possible a rationalization of the experimental results reported in the literature. A comparison of the precipitation conditions of different kinds of polyelectrolytes suggests a general process. Our experimental data are compared to the "ion-bridging" model based on short-range electrostatic attractions. By starting from the spinodal equation, predicted by this model, and using the limiting form of Manning's fractions of condensed counterions, analytical expressions of the precipitation conditions have been found in the three regimes. Experimental and theoretical results are in good agreement.