Yes, if the woman has reliable cycles and also pays attention to her cervical fluid.
When a woman ovulates, it intrinsically causes a rise in progesterone (this is known as the luteal phase). The presence of progesterone causes the body's basal temperature to rise by a few centigrade. Therefore, a slight raise in the BBT mid-cycle helps to confirm the beginning of the luteal phase, ie. ovulation. This 'temperature shift' pattern becomes predictable if the woman has consistent, normal cycles and so charting one's BBT (which is what the Lady Comp does for you) will give you a good idea of when to avoid sexual intercourse due to the foreseen ovulation. According to one study (1):
Basal temperature is an indirect measure of ovulation as a result of
the progesterone-induced temperature increase of ∼0.3 °C from the
follicular phase, when progesterone is low, to the luteal phase, when
progesterone peaks following ovulation . [...]∼75% of cycles had a
temperature rise within 3 days of the LH surge .
I did not see anywhere on the Lady comp site, however, that cervical mucus is also documented. Lady Comp takes into account the cycle length and basal body temperature (BBT), which is quite similar to the natural family planning method of the book Taking Back your Fertility by Toni Weschler (http://www.tcoyf.com/), except the latter also observes and documents cervical fluid changes (also known as cervical mucus). To have such a high success rate in preventing pregnancy as Lady comp claims, cervical mucus must also be accounted for because different phases of the menstrual cycle correlate to different phases of cervical mucus, and this factor determines whether or not the sperm can travel up the female reproductive tract to actually reach the ovulated egg. In a study done on bovine to determine fertility properties of this mucus in mammals (2):
Cervical mucus becomes more plentiful, watery, less viscous and easier
to traverse by spermatozoa in the follicular phase of the ovarian
cycle, as well as under oestrogen administration. On the contrary, in
the luteal phase of the cycle or under progesterone administration,
this mucus becomes scanty, opalescent, viscous and, consequently, not
favourable to sperm passage. Similarly, oviductal mucus becomes
progressively less viscous during a short period after ovulation
A close relationship between the rheological behaviour of bovine
vaginal ﬂuid obtained at oestrus and the molecular organization of its
structural elements has been shown. During the course of oestrus,
cervical mucus is able to achieve a dramatic reduction in its
mechanical barrier eﬀect while maintaining its three-dimensional
ﬁlamentous structure. This structure is related to a low consistency
index of samples, favouring sperm migration eﬃciency to the utmost.
Taken together, all these ﬁndings suggest that in mid-oestrus (...),
there seems only to be a short 2–4 h period (...) available for the
sperm to reach the uterus. Either side of this time interval, the
mucus is mechanically more hostile for the movement of sperm.
(1) Bedford J.L., Prior J.C., Hitchcock C.L., Barr S.I. Detecting evidence of luteal activity by least-squares quantitative basal temperature analysis against urinary progesterone metabolites and the effect of wake-time variability (2009) European Journal of Obstetrics Gynecology and Reproductive Biology, 146 (1) , pp. 76-80.
(2)Rutllant J, López-Béjar M, López-Gatius F. Ultrastructural and Rheological Properties of Bovine Vaginal Fluid and its Relation to Sperm Motility and Fertilization: a Review. Reproduction In Domestic Animals [serial online]. April 2005;40(2):79-86. Available from: Academic Search Premier, Ipswich, MA. Accessed January 4, 2013.