As also pointed out in the other answers, this will only cause problems if you are going to selectively consider the positive results where the null hypothesis is ruled out. This is why scientists write review articles where they consider previously published research results and try to develop a better understanding of the subject based on that. However, there then still remains a problem, which is due to the so-called "publication bias", i.e. scientists are more likely to write up an article about a positive result than on a negative result, also a paper on a negative result is more likely to get rejected for publication than a paper on a positive result.
Especially in fields where statistical test are very important will this be a big problem, the field of medicine is a notorious example. This is why it was made compulsory to register clinical trials before they are conducted (e.g. here). So, you must explain the set up, how the statistical analysis is going to be performed, etc. etc. before the trial gets underway. The leading medical journals will refuse to publish papers if the trials they report on where not registered.
Unfortunately, despite this measure, the system isn't working all that well.