  Spin ices, frustrated magnetic materials analogous to common water ice, are
exemplars of high frustration in three dimensions. Recent experimental studies
of the low-temperature properties of the paradigmatic Dy$_2$Ti$_2$O$_7$ spin
ice material, in particular whether the predicted transition to long-range
order occurs, raise questions as per the currently accepted microscopic model
of this system. In this work, we combine Monte Carlo simulations and mean-field
theory calculations to analyze data from magnetization, elastic neutron
scattering and specific heat measurements on Dy$_2$Ti$_2$O$_7$. We also
reconsider the possible importance of the nuclear specific heat, $C_{\rm nuc}$,
in Dy$_2$Ti$_2$O$_7$. We find that $C_{\rm nuc}$ is not entirely negligible
below a temperature $\sim 0.5$ K and must be taken into account in a
quantitative analysis of the calorimetric data of this compound below that
temperature. We find that small effective exchange interactions compete with
the magnetostatic dipolar interaction responsible for the main spin ice
phenomenology. This causes an unexpected "refrustration" of the long-range
order that would be expected from the incompletely self-screened dipolar
interaction and which positions the material at the boundary between two
competing classical long-range ordered ground states. This allows for the
manifestation of new physical low-temperature phenomena in Dy$_2$Ti$_2$O$_7$,
as exposed by recent specific heat measurements. We show that among the four
most likely causes for the observed upturn of the specific heat at low
temperature -- an exchange-induced transition to long-range order, quantum
non-Ising (transverse) terms in the effective spin Hamiltonian, the nuclear
hyperfine contribution and random disorder -- only the last appears to be
reasonably able to explain the calorimetric data.
