Some paramagnetic materials do become Ferromagnetic below the Curie temperature (or alternatively - ferromagnetic materials become paramagnetic when heated).
However, this is not a must: paramagnetism is due to the alignment of the magnetic moments along the magnetic field. The magnetic moments are effectively idnependent. Ferromagnetism, on the other hand, is due to the interaction between magnetic moments that aligns them in the same direction (which is not necessarily the direction of the external magnetic field - unless the latter is very strong). Higher temperature has potential for disrupting this interaction and converting a Ferromagnetic material into a paramagnetic one, but the converse is not true.
Magnetic susceptibility
Note that the difference between a paramagnet and a ferromagnet is not the strength of the magnetic response to the external field (i.e.,magnetic susceptibility), but the absence/presence of the spontaneous magnetisation - i.e., in Ferromagnet magnetization is present even in asence of an applied magnetic field, and persists, if the external field is not too strong.
The magnetic susceptibility of ferromagnets above the Curie temperature (i.e., in the paramagnetic phase) is described by Curie-Weiss law:
$$\chi\propto \frac{1}{T-T_C},$$
where $T_C$ is the temperature of the ferromagnetic transition. As we see the susceptibility increases with lowering temperature, until a material becomes ferromagnetic. More precisely, near the transition the exponent in the law may be different:
$$\chi\propto \frac{1}{(T-T_C)^\gamma}.$$
In paramagnetic materials, not exhibiting the ferromagnetic transition, this dependence is instead given by the Curie law:
$$\chi\propto \frac{1}{T},$$
which also predicts that the susceptibility increases with lowering temperature, but that it remains finite - the material does not become ferromagnetic.
