We calculate the number of dark matter particles that a neutron star accumulates over its lifetime as it rotates around the center of a galaxy, when the dark matter particle is a self-interacting boson but does not self-annihilate. We take into account dark matter interactions with baryonic matter and the time evolution of the dark matter sphere as it collapses within the neutron star. We show that dark matter self-interactions play an important role in the rapid accumulation of dark matter in the core of the neutron star. We consider the possibility of determining an exclusion region of the parameter space for dark matter mass and dark matter interaction cross sections based on the observation of old neutron stars with strong dark matter self-interactions. We show that for a dark matter density of $~10^3$ GeV/cm$^3$ and dark matter mass $m_chi$ less than approximately 10 GeV, there is a potential exclusion region for dark matter interactions with nucleons that is three orders of magnitude more stringent than without self-interactions. The potential exclusion region for dark matter self-interaction cross sections is many orders of magnitude stronger than the current Bullet Cluster limit. For example, for high dark matter density regions, we find that for $m_chisim 10$ GeV when the dark matter interaction cross section with the nucleons ranges from $sigma_{chi n}sim 10^{-52}$ cm$^2$ to $10^{-57}$ cm$^2$, the dark matter self-interaction cross section limit is $sigma_{chichi}< 10^{-33}$ cm$^2$, which is about ten orders of magnitude stronger than the Bullet Cluster limit.
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