In theory, the answer is yes, it can. There are 2 general cases (we assume that it's pure water):

First, the polymerization is not complete (chemical viewpoint). And, it continue reaction in aqueous environment when you immerse in water. Hence, the (molecular) weight is still enhanced.

Second, the 3D structure of polymer has ability to "hold" water molecular (hydrogel). This is the increase of polymer weight, not molecular weight.

However, to decrease the weight, it's another step. We cannot increase and decrease the weight in a step while conditions are constant (maybe it occur very quickly and we cannot observe, 0.0001s for increase and then decrease). An exception is a very long-time process; yes, it is possible in few months or years (I don't know why is it). Hence, after the increase of weight, to reduce it, we need to change: (1) temperature, (2) pH, (3) assistant steps such as microwave, photo-induce or electromagnetic field, etc.

I don't know exactly what situation you mention. But we can build a copolymer, it is a solution!

Thanks for you answer

Actually it's a copolymer; at first 24 hours its weight increased, but 24 hours later its weight decreased. The copolymer has been modificated with alkanolamine and sulfonic groups.

Herein, we need to define 3 factors: (1) the solubility of monomers, (2) the concentration ratio/distribution state of monomers and (3) the structure of copolymer (1D, 2D or network). The hydrophilic/hydrophobic of other groups is also important.

It's hard to explain with insufficient information. But, I guess that the copolymer you used contains at least one soluble polymer (i.e. PVA) with low concentration ratio, and have network (3D) structure. So, what's information/explanation/question you want to know? Please clarify your question!

Yes, you are right, I will be more specific

The copolymer is hydrophobic, but it was sulfonated and an alkanolamine was added; finally, the modificated copolymer was made membrane. The question is, what could hapen between sulfonic groups (on the sulfonated copolymer) and, amine and/or OH groups (on the alkanolamine) for explaining the behavior  its water absortion, I meant which reactions could hapen to explain the increase and decrease of copolymer weight when it is inmersed on water.

From your information, I try to give some explanations (we assume that no role of co-polymer):

1. The role of sulfonate group (-SO3H) is the retention of water disperability, prevent aggregation and eventually precipitation.

2. The role of amine just enhance the polarization of the system which improve the attraction of water molecular.

In summary, two groups can lead an increase of weight due to the retention of water molecular.

It's all. No reaction can happen. No more explanation

1. First step is increasing of weight due to swelling of polymer or polymer network.

2. Second step is decreasing of weight, phenomenon can be caused by wash out of oligomer or monomer from structure of polymer

The initial increase is obviously due to moisture absorption. The weight reduction may have been caused by the degradation of the polymer.

Si te refieres a la masa del polímero pudiera ser de acuerdo al grado de  higroscopia,  del polimero y entonces pudiera este absorber agua e hincharse y asi entonces aumentar su peso. Si es en relacion a la masa molecular es dependiendo si todavia no ha polimerizado por completo es posible que pueda darse un cambio en el peso molecular 

I've seen a similar result with fuel cell polymers that I know are completely stable in water and definitely clear of short-chain molecules (high MW, very low PDI, confirmed by GPC). It's not necessarily oligomer/monomer washing out of the system or degradation - presumably you're casting these as thin films, so there will be some remainder of your casting solvent that is trapped and (presumably) only becomes re-accessible when the polymer swells sufficiently.

There's another more fundamental angle - do you know the Tg(s) of your polymers, and have you / are you sure that you've fully exchanged it to the acid form? If you're above your Tg, you could be getting rearrangement and aggregation or crystallization that may enhance the hydrophobicity long-term. It's a very real possibility since you've got salt-like interactions that can form as the chains move around, and each sulfonate-amine bridge that forms is going to reduce the ability to form water channels. Sequential NMRs can help you sort it out - the water peak will shift around depending on hydration state.

Finally, there are a lot of very good publications on water management in proton exchange membranes (e.g. for fuel cells / electrolyzers) - there's a whole thesis out of my group (a result of quite a few ACS papers) that's really good at describing what I've mentioned, plus a few alternative possibilities, e.g. a change in interfacial hydrophobicity (around p120) causing a slow change in the vapour equilibrium:  http://summit.sfu.ca/system/files/iritems1/9977/etd5886.pdf . The PEM side is your best bet - lots of structural studies, good data. On the AEMFC side, Andrew Herring presents the only data I'd trust (it's a new field, modelling is very different from experimetnal data, etc.). Presumably groups that work on bipolar membranes have looked into the acid-base interactions at the interface, and that could directly relate to your work, but you may have to wait awhile for good studies on those.

 Thank you so much