Specular light is not modulated by the diffuse texture, so specularity can easily make a surface appear brighter than its texture at times, even in an LDR renderer.
But more generally, ordinary materials can't reflect more light than they receive, so (considering only diffuse for a moment) the reflected light will be no brighter than the incoming light, and if the incoming light is clamped to 1 then the reflected light will be no brighter than the texture.
In an HDR renderer the lights can have an arbitrary brightness, but that's not the whole story. Two keys to doing HDR properly are: setting the exposure properly based on the brightness of the scene and your aesthetic goals, and using a good tonemapping filter. If you want the image to look brighter you have to adjust the exposure to do so. Even if the lights are still in the [0, 1] range, you can adjust the exposure to make the image brighter. This is an artistic choice and it's perfectly permissible to set the exposure in such a way that well-lit objects end up brighter than their textures. The tonemapping filter allows you to alter the exposure without losing too much detail either in the bright or dark areas. It also makes bright colors ramp to white in a natural way as they get brighter, rather than simply being clamped, which can produce unwanted color shifts and loss of detail.
There's a good tonemapping equation on slide 140 of this talk (which is also highly worth reading, at least the first two sections on gamma correction and HDR, if you want to understand how to do physically based rendering correctly).