A teacher once told me that in Saxon times children under about five probably didn't wear clothes unless they were outside and it was cold. Is that true?

Not much seems to be known about the clothing of the time. From Dress in Anglo-Saxon England by Gale R. Owen-Crocker, published in 2010:

No contemporary historian has given us an account of how the Anglo-Saxon settlers dressed, and they themselves, being illiterate at this point, have left no documentary evidence.

The assumption that children wouldn't wear clothes might be plausible if you're thinking in terms of peasants and the very poor, unable to clothe them. Children would have lower priority because, well, they're children and they don't need to be clothed. The attitudes towards children, however, were very different than they are today. From "A Wryed Existence" by Mathew S. Kuefler in the Journal of Social History, on page 824, published in 1991:

The existence of child monks and nuns, called oblates, would seem to support the viewpoint of children as miniature adults, and bely a psychological appreciation of children.

~~From a quote by Barbara Greenleaf, on page 823 of the same article:~~

~~When they gave serious thought to children at all, people either conceived of them as miniature adults, or~~ ~~as peculiar, unformed animals.~~

Edit: As pointed out by /u/AnacreonInHeaven, the idea of children as "miniature adults" is likely false. I was taught that this was true, and after finding a few sources that supported the idea I decided to include it. However, it is likely an outdated belief about the time.

A larger discussion of the role and status of children can be found in chapter 13 of Children and Material Culture, edited by Joanna Sofaer Derevenski. From this idea that children had similar roles as adults, it may be safe to conclude that they dressed the same as well. This is supported by what has been found in graves and burial sites, though this would not be indicative of how they would be dressed while living.

I would say that your teacher was likely wrong, but in actuality we don't know. We can only make conjecture based on what we know of the role of children, and considering their roles were very similar to those of adults it may be an okay assumption that they dressed similarly too.

Edit: It was brought to my attention by /u/deadjawa that OP may have been referring to the Germanic Saxons, and not, as I assumed, the Saxon settlers in England. The point, as made in the previous paragraphs about Anglo-Saxon England, stands, however: we do not know due to the lack of documentation of the time. We mainly have inferences about what the culture was like since there is no direct evidence. As written in The History of the Medieval World by Susan Wise Bauer on page 389, published in 2010:

Charlemagne decreed that any "unbaptized Saxon who conceals himself among his people and refuses to seek baptism, but rather chooses to remain a pagan shall die." A Saxon who...indulged himself in the old Saxon rites instead of Christian worship, would be put to death. And any Saxon who did not observe Lent properly would be executed.

The conversion of the Saxons by Charlemagne was harsh, and so it's conceivable that Saxon culture was, as a result, changed drastically - which includes clothing. They likely switched to fashion considered more "acceptable" by Christian standards, and so it may be safe to say that it was also likely that children did not run around naked.