The half-life of a radioactive element means that, on average, half of the unstable nuclei will decay within one half-life. So if the half-life is 1 year, then after 3 years there will be (roughly) 1/8 of the original number of unstable nuclei in the sample. However, it is only an average, and once you get down to very small numbers the average only provides a very rough guide.

Iodine-131 has a half-life of about 8 days (actually more like 8.14).

1 g of iodine-131 contains about 4.6 x 10 ^{21} unstable nuclei.

After 8.14 days, there are about 2.3 x 10 ^{21} unstable nuclei.

After another 8.14 days, there are about 1.2 x 10 ^{21} unstable nuclei.

And....

So....

On....

After 70 half lives (about 570 days, or a year and a half), however, we might expect to be down to about 4 unstable nuclei, and another 8 days after that, two, and another 8 days after that, just one (note that because this is a random process by the time we get down to small numbers like this it's not really reasonable to assume that it'll all fall out neatly like this, but that's the general ballpark.

Now, we're down to one radioactive atom, and this is where it really becomes clear that to say that "It just keeps reducing to half after a fixed time" no longer applies:

A nucleus either decays or it doesn't... if you have 1 radioactive nucleus, there is a 50/50 chance of its decaying within one half-life, and if it doesn't then there's a 50/50 chance of its decaying in the following half-life, and so on. But we can be pretty confident it will eventually decay.

At this point, all the iodine-131 we started with has gone, we've got 1 g (well, near enough) of xenon-131 instead.

The answer to the original question, then, is "no", the half-life does *not* mean something can never decay away completely... although it may take a surprisingly long time.