Not completely correct. A better statement is that a test that rejects at 1% will have been rejected at 5%, 10%, etc. You shouldn't change the threshold for a test once it is set prior to the experiment.
The threshold at significance level x% is set such that x% of the distribution (by area under the density curve) lies beyond the threshold. Rejection means that the statistic you are checking against the distribution falls in this area. So if it falls in the 1% area, it will have also fallen in the 5% area, etc.