Do lilies have any diseases?

The worst pest of lilies rather than a disease are lily beetles which will strip off the leaves and eventually kill the plant if not controlled. Watch out for bright red beetles on the lilies and crush them if you do. Also look out for the little grubs which do most of the damage when they hatch out. Crush these too.
The lilies in your picture are probably suffering because they are in fairly small pots. In small pots they dry out quickly or can be overwatered. Lilies will do far better in a much larger and deeper pot or better still in open ground. If you can't plant in open ground why not make a Versailles tub out of some timber, line it with polythene and then plant your lilies into this (much cheaper option than buying a very large pot.)

Definitely no drying out Bertiefox. Overwatering? Hmm. Possibly. What is a Versailles tub? Is that a cube shaped one? I suppose when bulbs are in a pot and it is a fraction too small for them, it is like when a person wears a pair of shoes which are a size or two too small. Ouch!

Have your lillies flowered yet Jonathan? Mine seem to be growing onwards and upwards like a beanstalk - but no sign of a flower.

Sometimes lilies can suffer from viruses and fungal infections. When you water their pots, soil containing the spores can splash up onto the lower leaves, and transfer the infection. I grow all my lilies in pots, and mulch the surfaces with gravel to prevent the "splashing", Although the leaves look damaged, it shouldn't affect the flowering Jonathan.

My 'Regale' and 'African Queen' are blooming gloriously right now Ginellie, with great big blooms. But my 'Stargazer' is yet to flower, though the buds are getting quite big. The different varieties have their different flowering times. Are yours getting tall but without buds forming on them? ... I might try mulching them with some gravel Crazydi. I do tend to s-p-l-a-s-h things rather.

Mine are also Stargazer Jonathan and they do have buds so I'll keep waiting, thanks.

Just to answer Jonathan above, a Versailles tub is a largish wooden cube often with decorative round bits (balls) on the corner posts. I think they are a good idea as it's a cheap way to make a really big pot, as being a miser I get horrified at the cost of large clay or plastic pots nowadays!