Is there a period after you have had a cold when you cannot get a cold again?
The short answer is no, there is no minimum time between two colds. The reasons for this are different.

Mostly it has to do with the viruses which cause the common cold. There is a bunch of them causing what we call a cold, including rhinoviruses, coronacviruses, adenoviruses, human parainfluenza viruses, human respiratory syncytial virus, enteroviruses other than rhinoviruses and metapneumovirus. As much as 200 different virus serotypes are connected to this disease, alone 99 from the rhinoviridae. See the references 1-3 for more details.

Different serotypes means that two viruses of the same family (a rhinovirus for example) have different surface antigens on the virus and are not recognized by the same antibody. And this is what makes it difficult for the immune system. When it has learned to recognize and defeat one virus which causes a cold, it still has to learn how to do this with another (serotypically different) virus. So the adaptive immune system has to make highly specific antibodies every time it is faced with a new virus (which is of course not limited to cold viruses).

Once you have been infected with a virus and developed a sufficient immune reaction to develop antibodies, you are usually immune against this specific virus (with its specific serotype). Cold viruses live in communities, so when you move or travel, you will most likely encountered with different viruses, which your immune system doesn't know...

Finally there is another point which adds to the problem: Mutations of the viruses. The viruses are mutating constantly (RNA viruses at high rates) which changes their appearance and also their surface molecules (which are important for the recognition through the immune system). This leads to new serotypes, which are unknown to the immune system. See reference 4 for some more details.

References:


Understanding the symptoms of the common cold and influenza.
Sequencing and analyses of all known human rhinovirus genomes reveal
structure and evolution.
Common cold
Mutation rates among RNA viruses